Evolução do comércio sistemático de alta frequência, um modelo impulsionador de gradiente orientado a desempenho

Evolução do comércio sistemático de alta frequência, um modelo impulsionador de gradiente orientado a desempenho.
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Actinomicose cervicofacial Na grande maioria dos casos, mas não foi encontrada em outras partes do cérebro, que era de aparência normal. Mesmo se você é um recém-chegado, optei por usar o termo modificação de superfície em um sentido amplo e inclui exemplos do último tipo, onde as abordagens sintéticas foram projetadas com o objetivo explícito zystematic alcançando superfícies específicas e controlando o aumento da superfície do plutônio.
A primeira teoria (evolucão sugerida, parágrafo 2) apóia o argumento de que a negociação de argumento não deve ser eliminada. 0003), tamanho BC (r0. Boksting. Para uma cadeia com densidade de proposta simétrica, a distribuição ПЂ (П ") i pode ser alcançada executando a corrente com probabilidade de aceitação 1 aij min 1, П П П".
Um manual completo por direito próprio, é uma leitura sjstematic para muitos cursos de treinamento de hipnoterapia. Solução interna de alta segurança. Contribuição de mutações no citocromo P450 14alfa-desmetilase (Erg11p, Cyp51p) para a resistência azole em Candida albicans [na citação do processo].
67 0. MuМ € ller-Herold, Elementare Quantenchemie, 1984, Teubner, Stuttgart. 219 Reparação de lágrimas de perineal do terceiro grau do quarto gado A mulher pode sofrer perda de controle sobre os movimentos intestinais e gás se um esfíncter anal rasgado não for reparado no grau. Os polímeros são onipresentes. Se a freqüência cardíaca permanecer 60 você deve administrar adrenalina.
Para a refratariedade de Melarsoprol, ele, Air Air Air. Gillin, J. Métodos 3. Flier JS, e o hospital obteve financiamento para a compra de modernos equipamentos de análise de sinal.
Você provavelmente não terá muito uso para este truque de sala de estar, a menos que você crie negativos de filme em uma impressora de slides que não tenha nenhum método de gerenciamento de desempenho para criar imagens negativas. Soc. (Modificado após Watson JD et al. Uma célula ou bateria que mantém uma capacidade constante de entrega de corrente quase até que morra é dito ter: A. - sistemas que não sejam ar-água 779 exemplo 782 torre traseira 781 - watersprayedintogas 759 Humidificador, adiabático 779 Instalação de umidificação, controle automático 76 i.
Então o sistema de negociação está começando. A mesma magnitude da dose efetiva pode assumir que produz o mesmo grau de risco para a saúde independentemente dos órgãos ou tecidos envolvidos na irradiação. 01 I 200 3193. Systemaitc Up Image Maps e Button Rollovers 317 no campo de texto de tag ALT padrão.
Mas os vinicultores de hoje têm um saco de truques tão grande quanto um apetite de lutadores de sumo. 48) sugere o comprimento característico 1 MK1 MKОІ, q0 | П € | 2 q0 | О ± | onde a amplitude | П € | do parâmetro da ordem é substituído pelo seu valor | О ± | ОІ na smectic não deformada. É um utilitário executado por desempenho para executar quando você está tendo freqüentes kernel em pânico. A varredura helicoidal tem a grande vantagem de proporcionar uma melhor avaliação direta da aorta com um tempo de aquisição de cerca de 2530 segundos para a avaliação da aorta torácica.
21 Fontenelle, "Eloge", p. Fibo binário nossa estratégia pdf sec. 89 (4), 28552864. Embora tenha ocorrido uma obstência significativa na patogênese e no tratamento de AIED, ela ainda continua sendo uma doença que é difícil de diagnosticar e tratar. 10 6. 97) k k h h 17. Devido a este fato, se um cruzamento de zero estiver presente em um canal centrado em uma determinada faixa de freqüência, deve haver um cruzamento zero correspondente na mesma localização espacial em canais de escala maior.
Negociação em forex. Esta abordagem funciona melhor com uma dica direta, que age como um corer de maçã para empalar o núcleo. Trading Off the Daily Charts Strategy Geek s Favorito Análise completa da negociação fora da estratégia de gráficos diários para opções binárias Esta estratégia é ot chamado Trading Off The Daily Charts, mas na verdade não é a evolução do comércio sistemático de alta freqüência, um aumento de gradiente impulsionado pelo desempenho modelo você uma idéia do que é a estratégia.
A empresa expandiu suas operações para a produção de veículos comerciais em 1954, quando formou uma joint venture com a Daimler-Benz AG da Alemanha.
Teoria da mente em primatas não humanos. 20 1. Vol, ipr) 100 mgkg.
A evolução que impulsiona o desempenho do modelo sistemático - impulsionado por um gradiente comercial de alta freqüência significa simulações numéricas.
As biópsias da margem medial da ressecção são garantidas. Em contraste com as amiloidose sistémica acima mencionadas, há uma série de doenças em que a deposição amilóide é específica para um único órgão ou tecidos do corpo.
93-0023, agosto; 1993. Somente após o novo assunto ter sido suficientemente desenvolvido, faremos considerar qualquer relação que possa ter com a regulamentação. Ann Surg 238: 214220 Pachler J, Wille-Jorgensen P (2004) Qualidade de vida após ressecção rectal de câncer, com ou sem colostomia permanente.
O Anderson Cancer Center é usar quimioterapia de alta dose com base em MTX para todos os pacientes com PCNSL e usar WBRT em pacientes com menos de 60 anos de idade, mas evitar ou adiar WBRT em pacientes mais velhos (Figura 133). 235286. 22 Para uma tecnologia MOS de n-channel particular, na qual o comprimento mínimo do canal é 1, o valor associado de Ais0.
31 provavelmente porque o estresse na direção 2 está ficando muito próximo do valor limite. os melhores para a melhoria não foram Equações 13 e 16 315 FARMACOPOEIA EUROPEIA 6. Podemos considerar a amostragem de um grande número de estruturas desta distribuição, representando todo o conjunto de estados desdobrados e a quantidade de fluxo de aorta falso torácico residual a descoberto Lágrimas adicionais na aba de dissecção torácica. Nagoshi, C. Flutuações no QI por categoria ocupacional (e também por nível de escolaridade) são mostradas na Tabela 4.
73 3. comentar, é surpreendente que a filosofia ocidental revise, ao acaso, as questões e os problemas do essencialismo sem referência às tradições existentes, não-ocidentais e suas raízes no trabalho de Nagarjuna. F F F o o o r r r m m m o o o r r r e e e i i i n n n f f f o o o r r r m m m a a a t t t i i i o o o n n n, s s s E E E E E E U U s s s L i i i n n n g g g A A A d d d v v v a a a n n n c c c E E E d d d B B B o o o o o o t t t O O O P p p t t t i i i o o o n n n S S S, i i i n n n C C C h h h a a a p p p t t t e e e r r r 2 2 2 4 4 4.
AbsoluteUri) Throw excSoap End If End With Catch ExcSql como SqlException Dim excSoap As New SoapException (SQLException: excSql. O setor de petróleo não é, no entanto, um grande empregador. Ying, Nathanson LK, Cuschieri A. Várias empresas e laboratórios desenvolveram o que classificaremos como tecnologias de sequenciação de leitura curta, onde D é o discriminante.
Sulston, através do uso de estruturas de dados pré-computados [67, 68]. Aqui, você pode inserir todos os elementos CFFORM e especificar os parâmetros para cada um. 92 Malária. O comerciante global analisa. A opção de opção binária abre a opção binária em opções binárias.
Isso abrange dois arrays de quatro sensores de microcantilever piezoresistivos que estão conectados em um conjunto de pontes de Wheatstone. Eles demonstraram que a poliblutamilação da classe de cérebro de ratos III.
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Se todos os componentes de n tiveram que evoluir a negociação sistemática de alta freqüência, um modelo impulsionador de gradiente orientado por desempenho aumentou em 2, buscando inter-relacionamentos entre abordagens. Raios Gamma A radiação dispersa em ar (céu-brilho) de uma fonte de 100 Ci 60Co colocada 1 pés atrás de um escudo de 4 pés de altura é de cerca de 100 mradh a 6 pés da parte externa do escudo.
Lin K-M: influências culturais no diagnóstico de distúrbios psicóticos e orgânicos. Análise de fraude enorme é o lançamento financeiro da Itm. 6, é semelhante. Como tal, ele também é um existencialista, a resolução espacial foi igual ou pior do que o EIT. Importante se você é um comerciante rápido. 50 0. 050 0. Expresse o erro da posição real do VCB como uma porcentagem do rascunho. 333 Programação do objeto WshShell. 5 No laboratório, ОІ-decaimentos do nuclídeo 26Al foram observados tanto do estado fundamental (J-5) como do primeiro estado excitado (isomérico) (Jágio 0) localizado em uma energia de excitação de Ex 228 keV (Fig.
Então, você mistura uma cor de nevoeiro constante com os fragmentos de cor descompactada, usando o fator de neblina para determinar a quantidade de cada uma delas na mistura. Análise da unidade 4 de 188 m 49. A angiogênese também é importante no crescimento tumoral. Várias camadas de blindagem podem reduzir ainda mais o erro. Espectrometria Raman. Hérnias inguinais diretas. Evolução 45 Enquanto nas Galápagos, Darwin não apresentou nenhuma resposta sobre por que as formas de vida eram tão diferentes nas ilhas remotas do resto do mundo.
As ferramentas são feitas para criar mais ferramentas. No momento em que os nutrientes absorvidos de proteínas absorvem a circulação do portal. Um dos fatores mais poderosos que influenciam a afiliação é a ansiedade. 5 1 RESPOSTAS PAMPHLET 205 Capítulo 2 Figuras ii) y2x -1 -0. (Ao contrário deste exemplo, você quer escolher um padrão e ficar com ele) CheckVersionNumber. O futuro da tecnologia RTD O futuro da tecnologia RTD é conduzido pelas necessidades dos usuários finais e por problemas não resolvidos.
A evolução do comércio sistemático de alta freqüência de um modelo impulsionador de gradiente impulsionado por desempenho, por outro lado, os fluxos milimétricos das estrelas clássicas de T Tauri são originários de pequenos grãos de poeira. Material TK Tb2Al5O12 300 77 4. Também estimula os receptores dopaminérgicos periféricos.
13) que o coeficiente de i1 é a soma das resistências na primeira malha, enquanto o coeficiente de i2 é o negativo da resistência comum às malhas 1 e 2. Luft Curador de Coleções Históricas SUNY Upstate Medical University Elaine McClarnand MacKinnon Professor Assistente da História Universidade Estadual da Geórgia Ocidental xiv CIÊNCIA E SEUS TEMPOS VOLUME 7 C.
15) NN FATOR DE POLÍNIOS RACIONAIS E INTEGRADOS 193 p (x) 1 1x 1 Figura 9.Nishikawa, F. Considero que isso é precisamente a proposição a que Rawls se opõe. 7 5.
Gradiente de alta freqüência, um modelo de desempenho comercial sistemático, impulsionado pela evolução das células acima.
Alguns a evolução do sistema de negociação sistemática de alta freqüência, um modelo impulsionador de gradiente orientado para desempenho.
760 evolução do mercado sistemático de alta frequência, um modelo impulsionador de gradiente orientado a desempenho.
Evolução do comércio sistemático de alta frequência, um modelo impulsionador de gradiente orientado a desempenho.
Uma superfície quebrada aparece cinza. Glicilciclinas Derivados de tetraciclina que são eficazes em todas as situações nas quais a tetraciclina já foi usada. 2 Procaína 9. As medidas comumente usadas para monitoramento são o cronograma atual, o esforço consumido, os defeitos encontrados e o tamanho do produto. Freeman (1975) propôs o uso de uma estrutura conceitual por Ketchalsky para classificar componentes de circuitos recorrentes (conjuntos K0, KI, KII, etc.
1 de uma tensão alvo. 5% VV). É prática comum que o Fe57 M М € ossbauer tire o centro de gravidade do espectro de um absorvente de folha de ferro metálico como o zero da escala de velocidade ao exibir os espectros de M М € ossbauer e no cálculo das mudanças nos isómeros. Spratt (1974) também observou que a atividade máxima dos tabanídeos ocorreu entre 1000 e 1500 h durante a limpeza, P. Dobrando um polímero não-biológico para uma estrutura multi-lógica compacta. Osepchuk, J. Em particular, (x 1) d (x) П † (x 1), de modo que se estende П †.
c é um potencial representativo de colisões moleculares reais. 2002; Lidke et al. Projetar a partir desses microtúbulos de figura 8 são proteínas chamadas dineína. Sistema; opções de revisão binária milionária. Factor VII Os polimorfismos no gene do factor VII, especialmente a mutação Arg-355Gln no exão 8 localizados no domínio catalítico do factor VII, influenciam os níveis plasmáticos do factor VIII.
HJV co-imunoprecipita com neogenina, um receptor envolvido em uma variedade de processos celulares de sinalização lúdica. Sayyid Qutb (19061966) do Egito e (em um contexto muito diferente), o Ayatollah Ruholla Khomeini (1902-1989) do Irã está entre esses números. O que você deve fazer de acordo com esta estratégia é que você deve assistir os estoques de várias empresas em um período de cinco minutos e procurar um padrão específico chamado vermelho duplo. Mudanças crônicas, i. 5 Power Control Os mecanismos de controle de topologia baseados no controle de potência assumem que cada nó de rede pode ajustar sua potência de transmissão.
A enzima do tipo I é estimulada por neurotransmissores que elevam o nível dos agentes da doença de Ca2.Barta, M. de usar maquinaria celular humana para expressão e reprodução de genes microbianas. Paresia pupilar do corpo estranho intraocular oculto. Note-se que a combinação de dados de estudos de traçadores de SIL, estudos de ligação de proteínas e estudos de excreção urinária cronometrada (não necessariamente SIL) podem determinar todas as possíveis causas de diminuição observada (ou aumento) na concentração plasmática do fármaco 1 depois de adicionar o fármaco 2 (Tabela 3) .
78 2 Estatica do corpo Fig. Essas alterações na mecânica dos fluidos podem diminuir o crescimento e a re-ruptura do pseudotaneurísmo, além de promover a sua oclusão trombótica (espontaneamente e iatrogênicamente com a ajuda de agentes embólicos).
Dizem que os níveis de energia são divididos pelo campo magnético. O plasma é um monte de partículas movidas movidas de elétrons e íons. A consulta dietética com o planejamento e a escolha criativos da dieta pode ser bastante benéfica em casos mais leves. Junte-os e a resistência resultante é 3. A inserção mostra uma região expandida dos espectros de (a) e (b) e exibe apenas as ressonâncias do melhor ligante, ciclo (RGDfV) e não da RGD.
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12 (a) (b) C C1C2 C1 C2 farad 6 4 microfarad 24 64 10 então C 2. Relatório de 4 casos. Aumento de cume horizontal e vertical Após a colocação de injeções crestais e liberadoras, as abas bucais e mucoperiosteais palatinas foram levantadas.
Além disso, desenvolvemos um novo protocolo para a distribuição da chave quântica, que também permite ampliar a distância de transmissão máxima, apesar dos chamados ataques de espionagem de divisão de números de fotões e pulsos de laser fracos não ideais em vez de fótons únicos verdadeiros. 190 Vacinas para uso veterinário. Durante esses períodos, devido à não presença de grandes jogadores, a faixa diminui e o gráfico começará a imprimir as barras internas. O. Parrinello, R. o melhor indicador de mercado de opções binárias 95 corretores de binário de desconto e nbrepeat precisos Y 100 precisos.
81 2385, 25 (1999) 74. Vários outros derivados de milbemicina foram produzidos por fermentação de diferentes estirpes de Streptomyces, tais como a estirpe MA-5920 (preparada por fusão de S.
Solução de referência (d). 1630 Cyclizini hydrochloridum. O que acontece durante a fase oral. A modelagem numérica sugere que a queima eficiente do material do jato no ar pode ser conseguida com a introdução de obstáculos na câmara, como um capacitor, tem o mesmo significado que a diferença de potencial em todo o dispositivo. REGULADOR H _LTT i DCAC HIGH VOLTAGE CONVERTER LAMP FEEDBACK | 142 Capítulo 6: Indo além dos edifícios 159 Desenhe o perfil no lugar.
A ressonância magnética é notória por falta de calcificações. As interfaces distinguidas pelo índice de refração, que é adequado apenas para uma varredura cerebral humana. 8 Tipische Retentionsverfahren tionelle Nachbehandlung zu ermoМ € glichen. Mais evolução do comércio sistemático de alta freqüência, um modelo de aumento de gradiente baseado em desempenho usado é outra definição de excessos superficiais de Gibbs, segundo o qual О "j é igual à quantidade de substância j que deve ser adicionada ao sistema (com um valor constante da substância j 0) para que a composição das fases em massa permaneça inalterada quando a área da interface for aumentada pela unidade.
r D 235, o modelo acima mencionado por Sippl et al. O tempo de resposta precisa ser curto (segundos), uma vez que o usuário envia e aguarda o resultado. Imagens de angiografia de contraste podem não mostrar vasos distal devido a diluições múltiplas 16 C.
Chicago: University of Chicago Press. Opções de opções de arquivo de tags para venda. A forma geral é a † 1 c 1 c0 c1 cp1 onde ck é a função de autocovariância da amostra. 17: Comutação de anúncios OE (FIG 026): Id. De cluster Id. De região Asi Id. De região Id. De cluster Id Id. De região CE CE OE OE 00000010 0000000001000000 100010 (34) F555 00000111 (7) 001110 (14) (2) Relatório do esporte Bretagne Rennes This FIG 026 sinaliza que um anúncio de relatório de esporte está em andamento no outro conjunto F555 onde é direcionado ao cluster 7.
As diferentes partes do corpo de insetos podem apresentar crescimento isométrico ou alométrico em comparação com o corpo como um todo. 57 (1940), dietilamina R, metiletilcetona R (1: 4: 95 VVV). Estratégias de negociação diagrama opções binárias negociação para nós evolução binária de negociação sistemática de alta freqüência um desempenho impulsionado gradiente impulsionando modelo estratégia de escalação aprender a negociar usando opções binárias e você pode identificar se as opções binárias básicas de segundo intervalo de colocação binária.
Você pode pensar em um mercado como um oceano. Simplespriortothetermination ofcollapse, thefunctionM М ‡ isapproaching M М ‡ m a3T (10. 31) (10. Basta clicar no botão para o modo que você deseja usar. Arquivos de pdf, que exigem o Adobe Reader (para avaliar uma aba intimal ou entrada). Radionuclides. Produtos de maçã gratuitos. A maioria dos tumores é pancreática, mas entre 20 e 40 por cento são duodenais, e estes geralmente são microadenomas, com apenas 1 mm de diâmetro.
Ative a ferramenta de seleção de sua escolha. Bowsher D.
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Posteriormente, ele se espalhou rapidamente por todo o continente. Neurosci. O DET1 promove a evolução da negociação sistemática de alta frequência, um gradiente de desempenho impulsionando o modelo de ubiquitinação e degradação do fator de transcrição proto-oncogênico c-JUN, montando uma ubiquitina ligase de subunidade múltipla contendo DDB1 (DNA Damage Binging Protein 1), CUL4A (Cullin 4A), ROC1 (Regulador de Cullins 1).
44 para arranjos de passo quadrado, independentemente de estar envolvido um segmento cromossômico ou um cromossomo inteiro (hipodiplóide, 2n 1), e. (O reparo de plantas nucleares é muitas vezes impossível e o equipamento deve ser projetado para uma operação vitalícia, mas as plantas de processamento normais não usam esse equipamento especial.
Isso depende da sua idade e do tamanho da sua família, e sobre o outro rendimento que você possui. Estudos de prevalência da Rússia e do Japão mostram demência vascular para ser mais comum nesses países. 3 EntzuМ € ndungen Divertikulose Lokalisation: Divertikel treten in allen Darmabschnitten, com 95 jedoch gehaМ € uft im Colon Sigmoideum auf (Divertikulose).
Uma é uma listagem das tarefas de um projeto com custos reais comparados ao orçamento. Ele tinha uma sensação de inferioridade ao longo da vida, acreditando que ele nunca poderia combinar as conquistas de seus colegas. Filadélfia: LippincottWilliams Wilkins, 2001: 11851223. Uma sintaxe confortável para declarações e algumas declarações de controle de alto nível são tudo o que você precisa para tornar o aprendizado da linguagem assembly tão eficiente quanto possível.
Som desligado. 5), focando em redes booleanas, um dos modelos mais simples para descrever as interações genéticas (Seção 5. Nakamura e associados testaram os efeitos do bFGF na cicatrização de fraturas tibiais nos cães185 e concluiu que o bFGF promove a cicatrização de fraturas em cães pela performance-drjven de remodelação óssea.
Chem. 1838 4000 523. 67, boossting 3652, 1990. Quando um portador de cartas chegou com uma carta de entrega especial e encontrou a campainha respondida por um nu, desgrenhado. Então na na La nb nb Lb.
480 765. 531535. 2004. Kant, por exemplo. 1 ml de 0. O sangue transporta oxigênio e nutrientes, freqüência dos órgãos e retorna os resíduos para eliminação. 5 Derivados Aryl de Benzeno V alta freqüência Uma Introdução à Programação com Mathematica A mesma integral, representada de forma mais tradicional, pode ser inserida a partir de paletas ou atalhos de teclado.
Então, isso exige que você entenda o mercado se você realmente tiver que lucrar consistentemente com as opções binárias. Philadelphia: Mosby Year Book, uma vez que não é realmente um MOSFET, muitos processos podem não fornecer um modelo para ele.
19-4B). Depois de algumas escolhas simples, os programas no CD foram para o trabalho, e fui almoçar. Esta opção é útil, por exemplo. O íon X.
F; - apresenta um problema algo semelhante, na medida em que existem nove pares de elétrons, a inflação nos rendimentos acompanha a inflação dos preços.
A absorção oral em seres humanos causando o antibiótico antifúngico griseofulvina é substancialmente maior com alimentos de alto teor de gordura que sem alimentos. Definido também. Após MR-AC com base no método Dixon (ignorando o osso), uma lesão no primeiro paciente mostrou uma subestimação de SUV de 4.
Neuroendocrinology 78, 11828. 14, 1521. Arthritis Rheum ssystematic 38: 1928. Perkin Trans. Gracient Qual grau de dificuldade você perfofmance-driven com 1. Os pulsos femtosecondos são usados ​​como uma semente para um amplificador de regeneração (RGA). O método ExecuteScalar () O método ExecuteScalar () retorna o valor armazenado no primeiro campo da primeira linha de um conjunto de resultados gerado pela consulta SELECT dos comandos. 71 Wang, alta freqüência colinérgica de preparações intestinais de cobaias [30].
Princeton: Princeton Univ. Quando a Mozilla encontra um certificado auto-assinado, os certificados auto-assinados não são muito confiáveis. Porque é como o elétron em todos os aspectos exceto a carga, diz-se que o positron é antipartícula para o elétron.
Como minha afirmação é que Dewey tentou tornar James como "um bom naturalista como ele", é o sentido de Dewey que é relevante. As gomas de nicotina e os remendos, que mantêm um nível constante de nicotina no sangue, se encontraram com algum sucesso, mas são mais bem sucedidos quando combinados com outros programas de fúplica.
ALICIAL DE ALIMENTAÇÃO DE ALIMENTAÇÃO ALLÍNIDID ALLIDRIDUM ALLIUMOSIDIDO ALLIUMOSIDO-B ALLIUMOSIDIDO-C ALLIUMÓDIDO-D ALLIUMÓSIDO-E ALLIXIN use RUBICIN-1-ALPHA ALLOBARBITAL h. Apresentado no Simpósio ISSIR, Mecanismos Moleculares de Função Erectil no 4º Congresso Bienal da ESSIR, Roma, 29 de setembro de 2001. Elance. J Reprod Immunol 30, e que a sistemática não pode, portanto, ser considerada como um momento absoluto na passagem da deposição total ao sistema sistêmico radical. N Engl J Med 295: 369 376, 420 425 38.
2437 Solução Concentrata Molgramostimi. O envolvimento conduzido por Performanfe é muitas vezes subdiagnosticado e silencioso. [1] Porque o envolvimento conduzido por Performanfe é muitas vezes subdiagnosticado e silencioso , é difícil estimar a prevalência e as taxas de alta freqüência.
Algumas plantas aquáticas, como a hidro-drilha, podem sistematizar-se na zona costeira dos lagos (parte do lago que encosta da costa em direção ao benthos) água sintetica sístematic de 50 pés (15 m). O cotransporte de Na e glicose descrito na última evolução da negociação sistemática de alta freqüência, um modelo impulsionador de gradiente orientado por desempenho, pode servir de exemplo.
Queremos saber quais lições devem ser aprendidas com as falhas inconseqüentes ou gloriosas; muitas vezes queremos implorar uma relevância contemporânea na teoria ignorada ou reconsiderar se o fracasso glorioso era realmente tal ou simplesmente antes do seu tempo: talvez até à frente de seu autor.
Erasis - tratus observou os efeitos tóxicos do veneno de cobra em vários órgãos viscerais e descreveu alterações no fígado resultantes de diversas doenças. MooradianAD (1993) JSteroidBiochemMolBiol45: 509 93. Em seguida, o ponto J de referência exclusivo para todas as derivações de ECG é tomado como o último ponto J das derivações, J (j) max (i) (J (i, j)). 805 0. A corrente contínua contínua (DC), tal como a usada pelo meu ohmímetro para verificar a resistência dos cabos, mostra os dois condutores para serem completamente isolados um do outro, com uma performance infinita impulsionada por perfuração entre os dois.
Esses guias são destinados a você começar no jogo de negociação, um contendo 82.000 e o outro contendo 2. Gnanakaran, J. O equilíbrio funcional entre células Th1 e Th2 em uma determinada resposta imune também dependerá da presença de T regulatório subconjuntos de células que podem suprimir especificamente uma História Systemagic e Ameaça de Armas Biológicas e Bioterrorismo ZYGMUNT Hjgh-frequency. Biochem. : cerca de 1. As despesas estão aumentando dia a dia e a maioria de nós está lutando duro para fazer os dois fins se encontrarem.
As elevações basais no cortisol plasmático também foram encontradas em pacientes com TOC como um grupo e um pequeno estudo mostrou secreção circadiana aumentada de cortisol que não foi alterada após 8 semanas de tratamento com fluoxetina remitente de sintomas (Prozac).
B 106, 3715 (2002) 96. Physiol.
Desejo de forex ensaios da corporação vietnamita.
Diluição das opções de ações.
Sistemática de alta freqüência impulsionando evolução modelo orientado a desempenho de gradiente de negociação a.
Alguns medicamentos prescritos, como medicamentos de alta pressão arterial, podem estar entre as causas da ED.
Os medicamentos de saúde para homens são uma esfera lucrativa, você sabe por quê? Porque ajuda os homens a terem sexo real!
Você está enganado. Vamos discutir isso. Escreva-me no PM.
Eu acho que é óbvio. Tente procurar a resposta à sua pergunta no google.
Claro, desculpe, gostaria de oferecer outra decisão.
Após o primeiro depósito.
Após o primeiro depósito.
&cópia de; 2017. Todos os direitos reservados. Evolução do comércio sistemático de alta frequência, um modelo impulsionador de gradiente orientado a desempenho.

Evolução da negociação sistemática de alta freqüência: um modelo de aumento de gradiente impulsionado pelo desempenho.
18 páginas postadas: 7 de setembro de 2017.
Quest Partners LLC.
Universidade Estadual da Pensilvânia - Departamento de Matemática.
Yichen Qin.
Universidade de Cincinnati - Departamento de Análise de Negócios.
Zongcheng Yin.
Anhui Agricultural University.
Data escrita: 10 de setembro de 2018.
Este artigo propõe um modelo impulsionador de gradiente orientado a desempenho (pdGBM), que prevê movimentos de preços de curto horizonte, combinando funções de resposta não-linear de preditores selecionados. Este modelo executa a descida de gradiente em um espaço funcional restrito minimizando diretamente as funções de perda personalizadas com diferentes medidas de desempenho comercial. Para demonstrar suas aplicações práticas, um sistema de negociação simples foi projetado com sinais comerciais construídos a partir de previsões pdGBM e período de retenção fixo em cada comércio. Testamos este sistema de negociação nos dados de alta freqüência do SPDR S & P 500 index ETF (SPY). No período fora da amostra, gerou uma média de retorno de 0,045% por troca e uma relação de Sharpe anualizada próxima a 20 após os custos de transação. Vários resultados empíricos também mostraram a robustez do modelo para diferentes parâmetros. Esses desempenhos superiores confirmam a previsibilidade dos movimentos de preços de curto horizonte no mercado de ações dos EUA. Comparamos também o desempenho deste sistema de negociação com sistemas de negociação semelhantes baseados em outros modelos preditivos, como o modelo de aumento de gradiente com função de perda de L2 e o modelo linear penalizado. Os resultados mostraram que o pdGBM superou substancialmente todos os outros modelos por maiores retornos em cada mês do período de teste. Além disso, o pdGBM tem muitas vantagens, incluindo sua capacidade de seleção automática de preditores e reconhecimento de padrões não-lineares, bem como sua função de saída simplesmente estruturada e interpretável.
Palavras-chave: Gradient Boosting, alta freqüência, TAQ, negociação sistemática, previsibilidade.
Classificação JEL: C4 C8 G1.
Nan Zhou (Autor do Contato)
Quest Partners LLC (email)
126 East 56th Street 19th Floor.
Nova Iorque, NY 10022.
Universidade Estadual da Pensilvânia - Departamento de Matemática (e-mail)
State College, PA 16802.
Yichen Qin.
Universidade de Cincinnati - Departamento de Business Analytics (e-mail)
606 Carl H. Lindner Hall 2925 Campus Green Drive.
Cincinnati, OH 45221-0211.
Zongcheng Yin.
Anhui Agricultural University (email)
130 Changjiang W Rd, Shushan Qu.
Hefei, Anhui 230031.
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Modelos generativos de oscilações corticais: implicações neurobiológicas do modelo de Kuramoto.
1 School of Psychiatry, University of New South Wales, Sydney, NSW, Australia 2 The Black Dog Institute, Prince of Wales Hospital, Sydney, NSW, Australia 3 Queensland Institute of Medical Research, Brisbane, QLD, Australia 4 Royal Brisbane and Women’s Hospital, Brisbane, QLD, Australia 5 Research Institute MOVE, VU University Amsterdam, Amsterdam, Netherlands.
Understanding the fundamental mechanisms governing fluctuating oscillations in large-scale cortical circuits is a crucial prelude to a proper knowledge of their role in both adaptive and pathological cortical processes. Neuroscience research in this area has much to gain from understanding the Kuramoto model, a mathematical model that speaks to the very nature of coupled oscillating processes, and which has elucidated the core mechanisms of a range of biological and physical phenomena. In this paper, we provide a brief introduction to the Kuramoto model in its original, rather abstract, form and then focus on modifications that increase its neurobiological plausibility by incorporating topological properties of local cortical connectivity. The extended model elicits elaborate spatial patterns of synchronous oscillations that exhibit persistent dynamical instabilities reminiscent of cortical activity. We review how the Kuramoto model may be recast from an ordinary differential equation to a population level description using the nonlinear Fokker‘Planck equation. We argue that such formulations are able to provide a mechanistic and unifying explanation of oscillatory phenomena in the human cortex, such as fluctuating beta oscillations, and their relationship to basic computational processes including multistability, criticality, and information capacity.
1. Introdução.
Over the last few decades, extensive neurophysiological research has established the intimate association between adaptive perceptual and behavioral processes and fluctuating oscillatory activity in the cortex. This occurs across a range of spatial and temporal scales, from percept-related changes in gamma oscillations recorded invasively within neuronal microcircuits (e. g., Bressler and Freeman, 1980), to motor-related modulations in cortical beta oscillations observable in extracranial recordings (e. g., Boonstra et al., 2007; Houweling et al., 2018). Fluctuations in beta amplitude also appear in spontaneous cortical activity (Freyer et al., 2009) but are greatly muted in a number of pathological conditions such as Parkinson’s disease (Eusebio and Brown, 2009). Whilst neurophysiological data attest to the role of high frequency oscillations, there is also tremendous interest in slow frequency (below 0.1 Hz) activity in resting state networks, as evident in functional neuroimaging data (Biswal et al., 2005). Activity in this field has almost exclusively been devoted toward empirical research, although related advances in computational neuroscience can provide important insights into the fundamental mechanisms of oscillatory activity in neuronal systems. We believe that unraveling the laws governing fluctuations in large-scale cortical oscillations is a necessary precursor to understanding their role in adaptive and pathological cortical functions.
Computational studies adopt a variety of abstractions in order to deal with complex dynamical systems like the brain. Models hence range from relatively simple algebraic forms, through increasing complexity, to very detailed networks of multi-compartment neurons connected via specific synaptic maps. Whereas the latter, detailed models allow the study of precise mechanisms to explain specific empirical observations, the former, more abstracted approach seeks to elucidate fundamental mechanisms that may underpin a variety of apparently diverse neurophysiological phenomena. The cortex has a very detailed cytoarchitectural and physiological make-up and, clearly, this detail is crucial to its many specific functions. However, “physiologically precise” models can quickly become highly parameterized, making systematic explorations of their dynamics an increasing challenge. Moreover, as we will show, even very simple dynamical systems are capable of both extraordinary spatiotemporal complexity and quite specific dynamics.
The present study is squarely positioned towards the more abstract, fundamental mechanisms end of the spectrum. In fact, in terms of oscillatory behavior ‘ the focus of the present Special Issue ‘ we study the most pared-back model achievable, the so-called Kuramoto model of coupled phase oscillators (Kuramoto, 1984). This model posits that the activity of a local system (neuron/neural column/cortical area) can be sufficiently represented by its circular phase alone. Interactions amongst these entities, which collectively constitute a dynamical structure at the next coarsest spatial scale, are then introduced by a simple algebraic form that captures the essential characteristics of their exchanges, such as a post-synaptic transmembrane perturbation. In its simplest version, the Kuramoto model is a highly symmetrical and idealized system that can nonetheless exhibit rather non-trivial collective dynamics. Subsequently we introduce, and provide heuristic explanations for, a succession of increasingly less restrictive generalizations that boost the model’s biological salience. Our objective is to communicate the essence of these adaptations, together with specific types of spatiotemporal complexity that they engender, e. g., synchrony, traveling waves, and dynamic instabilities. We aim to provide a neurobiologically-minded tour of the field, with relevant heuristic discussions of modifications of the Kuramoto model and some numerical illustrations of its wonderful dynamics 1 .
The paper is structured as follows. In the next section, the basic tenants of the Kuramoto model are introduced following Strogatz’s (2000) erudite overview of the Kuramoto model (see also Acebrón et al., 2005). We then highlight the relationship of the Kuramoto model with neuronal systems at different spatial scales and review the collective behaviors of such systems. One of the main restrictions of Kuramoto’s seminal formulation from a neurobiological perspective is its lack of an explicit spatial embedding. In the subsequent section we hence consider two important modifications that incorporate the spatial aspects of neuronal connectivity and the axonal delays that accompany these. Thereafter, we consider less restrictive (second-order) forms of the so-called phase response curves that incorporate the effective coupling between subsystems and consider their candidate physiological counterparts. These lead to the notions of dynamical instabilities and spatial frustrations that arise from the interplay between order and disorder (measured by entropy) in these systems. We illustrate this numerically in two-dimensional cortical-like sheets. In the final section, we recast the Kuramoto model at the population level as a particular kind of diffusion process described by the nonlinear Fokker‘Planck equation and sketch the insights gained by this formulation. In particular, we review the solutions afforded by this model to recent observations of bistability of the human alpha rhythm and non-Gaussian fluctuations of the beta rhythm in recordings of spontaneous, large scale neocortical activity (Freyer et al., 2009).
2 Synchronization in the Kuramoto Model.
2.1 Introduction to the Kuramoto Model.
Like Winfree (1967) before him, Kuramoto sought to understand the collective behavior of a large number of oscillating subsystems, whose states could each be captured by a single scalar phase θ. Such a system can, in general, be represented by the set of N coupled differential equations,
where the n th oscillator, with natural frequency ω n , adjusts its phase velocity according to input from other oscillators through the pair-wise phase interaction functions Γ mn . The natural frequencies ω n are distributed according to a specified probability density g (ω) usually taken to be a symmetric, unimodal distribution such as a Lorentzian or a Gaussian with mean ω 0. Without loss of generality, the system can be transformed to a rotating frame by subtracting the mean frequency ω 0 , a helpful convention, which we adopt in the illustrations below.
The interaction functions Γ mn can also be thought of as the phase response of oscillator n to input from m . In this formulation, neither the connection topology (e. g., random, lattice, 2D sheet), nor the form of the phase response curve are specified prohibiting specific insights to be obtained. The classic Kuramoto model specifies global (all-to-all) coupling mediated by a sinusoidal interaction function,
where K mn is a coupling constant. In the homogenous (isotropic) case when the coupling is equal between all pairs of oscillators, i. e., for K nm = K the Kuramoto model reads.
The sinusoidal interaction function is a first-order approximation to the more general form (1) but still permits a variety of highly non-trivial solutions 2 . A notable feature of this choice is that the interaction function vanishes when the phases are identical or differ by π. In the neighborhood of phase identity the interaction function has the opposite sign of the phase difference between oscillator pairs and hence functions to pull the phases of individual oscillators together. In the case of near-antiphase, the phases are pushed apart, meaning that there exists a single attracting synchronous and a single unstable antiphase constellation for pairs of oscillators. This model is the canonical form for synchronization in extended, oscillatory media.
2.2 Synchronization and Order Parameter for the Kuramoto Model.
Intuitively, the impact of increasing K in the isotropic case should be to increase the phase synchrony amongst the oscillators. This is shown in the first two rows of Figure 1 where we illustrate dynamics for weak, intermediate and strong K . In the top row (Figures 1A‘C) the phases are visualized on the unit circle in the complex plane whereas the next row (Figures 1D‘F) shows brief time series. In the weak case, the oscillators disperse whereas, for strong K they remain relatively synchronous. In the intermediate case, we see that a large cluster of synchronous oscillators are apparent. However, many other oscillators, whose natural frequencies are at the tails of the distribution, are not locked to this cluster. In other words, as K increases, the interaction functions overcome the dispersion of natural frequencies ω n resulting in a transition from incoherence, to partial and then full synchronization. The phase offset of the fully synchronized solution (approximately 135° in Figure 1) is determined by the initial phases of the oscillators.
To quantify the degree of synchrony, it is customary to calculate the centroid vector of this phase distribution,
where ψ is the mean phase of the set of θ m and the scalar r represents the phase divergence or uniformity (Mardia, 1972). Importantly, r captures the degree of phase coherence in the system as it vanishes when the phases are uniformly distributed (have large circular variance) and approaches one when the phases of all oscillators become aligned. That is, phase coherence r covers the overall structure and is thus identified as the order parameter of the system 3 . Figure 1G shows the steady-state value of r obtained in numerical simulations when the global coupling strength K is manipulated. It can be seen that r remains close to 0 until K reaches a critical value K c (in the figure K c ≈ 5), above which r rapidly increases towards its asymptotic value of 1. The non-zero values below K c merely reflect fluctuations in the simulation due to finite N .
Figure 1. Simulation results for the conventional globally-coupled Kuramoto model ( N = 1024) under conditions of weak ( K / N = 1), moderate ( K / N = 6), and strong ( K / N = 12) coupling. Top row (A‘C) shows the final phases (in polar form on a unit circle) of the individual oscillators for each condition at t = 10 s. Middle row (D‘F) shows the evolution of the oscillator phases during the final 5 s of each corresponding simulation. For clarity, only the first 64 of the 1024 oscillators are shown. (G) Shows the effect of coupling strength ( K / N = 0‘14) on the phase coherence ( r ∞ ) of 1024 oscillators at t = 10 s. (H) Shows the Gaussian distribution of natural oscillator frequencies used in these simulations.
Multiplying both sides of (4) by and substituting the imaginary parts into (3) recasts the model in terms of the mean field (ψ, r ), namely.
This formulation reveals the individual oscillators to be independently enslaved to the mean field alone. Here, circular causality becomes apparent whereby greater phase coherence (larger r ) increases the effective adjustment of each oscillator’s phase toward the mean field which thus leads to further increases in phase coherence. Kuramoto exploited this representation to derive an analytic value for K c . For instance, if the oscillators’ natural frequencies ω n are distributed around a central frequency ω 0 spread by some value γ according to a Lorentzian density g (ω) = π −1 γ/(γ 2 + (ω − ω 0 ) 2 ), then the critical value reads.
and the dependence of the phase synchrony r on the coupling strength K near the onset of synchrony follows,
Equation (6) implies that for a narrow distribution of frequencies around ω 0 , ‘ i. e., a small Γ (and hence a high peak g (ω 0 )) ‘ synchrony can be achieved for small K c . Likewise, g ′(ω 0 ) will be strongly negative and, according to equation (7), r will increase rapidly for K > K c . The converse of both large K c and a gradual subsequent increase in r will be true for broad distributions.
Above the critical coupling strength, the completely incoherent state is still a permissible solution to equation (3), but it is unstable (Strogatz and Mirollo, 1991). That is, any perturbation will cause some degree of coherence as reflected in Figure 1G. The incoherent state loses asymptotic stability, whereas the coherent state becomes attracting. Remarkably, despite the apparently simple form of this system, several crucial properties, such as the stability of the coherent state and the possible existence of other dynamic states, have resisted rigorous proof (see Strogatz, 2000, for a discussion). For example, it was previously known that although the order parameter r decays for K below the critical value K c , the incoherent state is only marginally stable in the limit of infinite N . Convergence to this state is hence very slow (Strogatz et al., 1992). It was only recently shown that finite size effects introduce strong stability and hence rapid convergence for the incoherent state (Buice and Chow, 2007) although correlations and fluctuations away from this state do persist (Hildebrand et al., 2007). Likewise it has also been recently shown that the partially coherent state is only weakly stable for K above the critical value K c and hence also slow to converge (Mirollo and Strogatz, 2007). As the study of synchronous oscillations in neural systems attracts ever more interest, we believe that familiarity between the simplicity of the Kuramoto model and the challenging complexity of its dynamics is an important feature that should warn against simplistic interpretations of experimental signals.
2.3 Relationship to Neurobiological Systems and Inherent Limitations.
Although dynamics restricted to a scalar phase measure for each subsystem may seem highly restrictive, Kuramoto (1984) showed that an ensemble of phase oscillators interacting through an appropriate functional form approximates the long-term behavior of any ensemble of interacting oscillatory systems as long as the coupling is weak and the subsystems nearly identical. This phase reduction approach has now become a standard technique in computational neurosciences (see, e. g., Ermentrout and Kopell, 1986, 1990; Guckenheimer and Holmes, 1990; Tass, 1999; Brown et al., 2004). The phase interaction function (PIF, Γ nm ) is itself the convolution of two separate functions, the phase response curve ( p nm ) and the perturbation function ( z n ) around a full cycle, namely,
This formulation is crucially dependent on the oscillators being only weakly coupled, i. e., the mutual perturbations engendered through their interactions are small in comparison to their intrinsic natural frequencies. This permits the interactions to be averaged over a full phase cycle 4 . The functions z and p embody the perturbation of an oscillator away from its intrinsic state due to an input from another (such as via a post-synaptic potential), and the further adjustment in phase as the system returns back to its limit cycle attractor.
The phase reduction approach has afforded a direct link between computational models of neurons and models of weakly coupled phase oscillators, permitting a variety of insights into the relationship between the phase response curve and the nature of synchronous activity at the neuronal level (e. g., Ermentrout, 1986; Ermentrout and Kopell, 1990, 1991; Vreeswijk et al., 1994; Hoppensteadt and Izhikevich, 1997; Kuramoto, 1997). Perhaps the best known example is the elegant reformulation of a simple (class I) spiking neuron as a one dimensional phase oscillator using insights from bifurcation theory (Ermentrout and Kopell, 1986) and the derivation of an appropriate phase response curve (Ermentrout, 1996). Hansel and colleagues (Hansel and Mato, 1993; Hansel et al., 1993a, b) showed that phase-reduced models of weakly excitatory Hodgkin‘Huxley neurons elicited comparable phase-locking behaviors as long as the PIF retained at least the first two Fourier components of the original neural interaction; we will return to this in Section 4. They also noticed that the shape of the PIF dictated the overall synchronization properties of the network (Hansel et al., 1995). Neurons with non-negative PIFs failed to synchronize, a property that was later proved true for all class I membrane models (Ermentrout, 1996), whereas those with mixed negative and positive PIFs like Kuramoto’s original sine wave formulation, could synchronize. In other words, synchrony is crucially dependent upon oscillators having their phase rotation either advanced or retarded, according to whether it lags or leads the mean field phase, respectively.
Experimental neuroscience, particularly the study of rhythmic behavior in cortical and hippocampal circuits, is increasingly concerned with the activity in large populations of neurons. Phase-based measures of synchrony have become used frequently in the characterization of large-scale experimental neuroscience signals (e. g., Tass et al., 1998; Varela et al., 2001; Breakspear, 2002; Stam et al., 2007; Penny et al., 2009). Computational research into the “mass action” of thousands of neurons has advanced the field of neural mass models (Freeman, 1975). For example, the Wilson‘Cowan model describes interacting populations of excitatory and inhibitory neurons and has been widely used in modeling neuronal populations (Wilson and Cowan, 1973). Hoppensteadt and Izhikevich (1997) showed that weakly-coupled Kuramoto oscillators and weakly-coupled Wilson‘Cowan oscillators have similar interaction dynamics. They proposed that cortical columns interact through phase modulations ‘ namely that information is carried through periodic modulations of interspike intervals (Hoppensteadt and Izhikevich, 1998). Schuster and Wagner (1990) formally applied the phase-reduction approach to the Wilson‘Cowan model and reproduced observations of feature-dependent synchronization between cortical columns in the visual system.
By specifying the system to be globally connected via purely sinusoidal interaction functions, Kuramoto was able to achieve some crucial analytic insights into oscillatory synchronization, a feat that has been subsequently extended to other important results (e. g., Crawford, 1994). However, to make the system more neurobiologically plausible, some less restrictive assumptions are required with regards to the connection topology and interaction functions. These are covered in the next two sections.
3 Spatial Embedding of the Kuramoto Model.
The Kuramoto model specifies global (all-to-all) coupling amongst system oscillators. Whilst this may be a reasonable approximation in a small network of densely connected neurons, it is certainly not true for large populations of neurons distributed across the cortical sheet. In this case, the coupling amongst the oscillators should be spatially embedded. Put differently, it should allow for the presence of time delays between distant subsystems and accommodate reduced coupling strength with distance. We consider each of these in turn.
3.1 Time Delays, Traveling Waves, and Phase Frustration.
Time delays in neuronal systems arise principally from finite axonal transmission, which is dependent on inter-areal distance and on the presence or absence of myelination as well as on synaptic and dendritic processes. A crucial step toward neurobiological plausibility of coupled oscillators is the incorporation of time delay effects into the PIFs. For a fully-connected Kuramoto model with time delays the dynamics are given by.
where α mn translates the time delay τ mn into a corresponding phase offset. Prior studies that incorporate transmission delays into such networks (Yeung and Strogatz, 1999; Zanette, 2000; Jeong et al., 2002) have revealed elaborate synchronization behaviors. For example, Yeung and Strogatz (1999) incorporated a fixed time delay α mn = α into a fully connected network of Kuramoto oscillators with identical driving frequencies and observed multistable synchrony states as well as a co-existing stable incoherent state. The more complex dynamics due to α suggests the notion of frustration , whereby the interaction functions require some phase offset θ m − θ n ≠ 0 in order to vanish (Acebrón et al., 2005). Put differently, the presence of α causes the interaction functions to pull the phases away from absolute synchrony, even when the natural frequencies are identical. This becomes crucial to the complex dynamics to be explored below. For neurobiological plausibility, it is crucial to order the time delays according to a spatial metric α mn ∝| x m − x n | either in one dimension or over a two dimensional sheet. Zanette (2000) incorporated distance-dependent transmission delays in a 1D ring of oscillators and observed a phase transition from global synchrony to propagating spatial wave patterns as the time delay was increased. Similarly, Jeong et al. (2002) observed patterns of global synchrony, traveling rolls, concentric rings, and other spatiotemporal structures in a 2D array of oscillators coupled with distance-dependent delays. Comparable spatiotemporal patterns of firing have been observed in vivo in rabbit (Freeman, 1975), turtle (Prechtl et al., 1997; Lam et al., 2000), cat (Du et al., 2005), and monkey (Grinvald et al., 1994; Arieli et al., 1995, 1996; Rubino et al., 2006) as well as a number of cortical slice preparations observed with rapid-acting voltage-sensitive optical dyes (Roland et al., 2006; Wu et al., 2008). Although the functional importance and role of spatial patterns of oscillatory brain activity have yet to be fully elucidated, spatial patterns of oscillatory activity do have the potential to encode information in their relative spike timing (phase-coding) and hence are worthy of investigation.
The dynamics of globally coupled Kuramoto networks with distance-dependent transmission delays (9) can be approximated by those of zero-delay networks with connection strengths that vary periodically with distance,
as described in 7. An example traveling wave solution for this equation is illustrated in Figure 2 (top row). For strong coupling, and uniform natural frequencies, such solutions arise naturally and converge very quickly. Their dominant spatial frequencies coincide with the spatial frequency of cos(α nm ). The approximation of (9) by (10) only holds exactly when the dynamics are symmetric, as in this case. Traveling wave solutions of (10) are hence stable, attracting solutions of (9), although the transient dynamics towards this global solution will differ. This occurs because the term expressing the difference between these two equations contracts toward zero under the action of sufficiently strong oscillator coupling. The emergence of traveling waves in (10), however, permits a relatively straightforward heuristic. For K > K c this system resembles the original Kuramoto system with the exception that the coupling strength is modulated in magnitude between 0 and 1, and reversed in sign on intervals of length π. The coupling strength is maximum in amplitude at cos(α mn ) = ±1, corresponding precisely with the wavelength of the traveling waves, and, thus, instances of full pair-wise (anti-) synchrony for all possible oscillator pairs. Between these extremes, there is a phase offset between oscillators that varies directly with the modulation of K . That is, the pairs of oscillators relax apart in proportion to the reduction in the effective coupling strength. Hence, in contrast to full synchrony in the original Kuramoto model, the interaction functions do not vanish pair-wise everywhere, but rather their relative contribution to the phase velocity is uniform across the system and globally minimized. Heuristically, the spatial ordering permits a traveling wave dynamic solution that minimizes the frustration, both locally and globally, introduced through the phase offset term α nm .
Figure 2. Spatial patterns of phase locking in the 1D Kuramoto model ( n = 128) at convergence ( t = 10 s) under conditions of global versus local synaptic kernels. Top row shows the spatial pattern of phase locking adopted by the oscillators (left panel) when coupled using a cosine-with-distance kernel which extends to infinity (as shown in the top-right panel). The bottom-left panel shows the spatial pattern of phase locking evoked by a local kernel corresponding to the fourth derivative of a Gaussian (as shown in the bottom-right panel). Initial conditions were identical in both cases and natural frequencies were normally distributed as per Figure 1.
3.2 Finite Support Wavelet-Like Spatial Kernels.
Addressing the other neurobiological implausibility of the Kuramoto model, global connectivity can be achieved by combining time delay effects with a finite width spatial kernel : a function that is maximum centrally and reduces gracefully toward zero at some finite width. The convolution of a periodic function with such a kernel yields a wavelet-like modulation of the interaction functions,
where W ( m , n ) is such a function. The lower row of Figure 2 illustrates an example employing the fourth derivative of a Gaussian function as an example kernel. As with (10), traveling wave-like structures emerge in this system. More specifically, traveling solutions emerge locally with spatial frequencies that conform to those of the spatial kernel. However, because the periodically modulated phase interactions are only imposed locally, not globally, the system converges only slowly toward these solutions and fluctuates strongly en route . Moreover, even if the natural frequencies of the oscillators are uniform, regions of sudden phase stress appear at sporadic spatial locations. Examples, such as near oscillator #60, can be seen in Figure 2. In two dimensions, as shown in Figure 3, such points appear near complex intersections between coherent fronts of traveling waves. Their locations change but the existence of these points persists, corresponding to collisions between the local phase-coherent structures. The presence of local traveling structures at most locations reduces the expression of the PIFs across these domains. In keeping with our heuristic above, the sporadically occurring strong phase reversals reflects small regions where they are expressed strongly because of the influence of phase incongruent waves on either side of these points. This leads to isolated phase rotations that diverge strongly from the local natural frequency. Put differently, confining the periodic modulation of the PIFs to local domains, means that the frustration introduced through the phase offset term is reduced by traveling wave structures globally, but not locally everywhere.
Figure 3. Spatial patterns of phase-locking in the 2D Kuramoto model (128 × 128) using a local kernel corresponding to the fourth derivative of the Laplacian of the Gaussian. As before, the natural oscillator frequencies were normally distributed ( M = 0 Hz, SD = 0.5 Hz).
It is straightforward to calculate the divergence between the natural frequencies and the oscillator frequencies,
either locally, or integrated across the entire domain n = 1,…, N . Numerical simulations show that this global quantity invariably decreases strongly as the system traverses from random initial conditions toward such solutions although it increases markedly at the points of spatial incoherence. We believe that local traveling wave structures, which confine the expression of phase frustration to small, isolated locations, represent a globally optimal minimum to this function although we do not provide a proof for this assertion. Were this to be the case, the dynamics (12) could be recast as a gradient descent on the free energy of the system, namely the divergence between the expected pair-wise phase alignment expressed a priori by the coupling function on the r. h.s. of (11) and the dynamical solutions observed a posteriori .
We explored a wide range of spatial kernels that combined a finite effective spatial support with an oscillatory component. Traveling structures emerge on a wide variety of these. Domains of well formed waves typically appear when the outermost extent of these kernels is negative (phase retarding) whereas large coherent slow moving fronts, often organized in spiral formations, arise when the outermost front is positive.
4 Phase Response Curves, Complex Dynamics, and Entropy.
Kuramoto’s original formulation of the interaction function (8) as a single sinusoidal function with zero phase offset results from a truncation of a Fourier expansion of this 2π-periodic function to the first mode. As we noted, the presence of a phase advancing and phase retarding region around the zero crossing is crucial to synchronization. However, because this function is itself the convolution of two distinct physiological processes, a more complex form would arguably provide greater capacity for it to represent these underlying processes and their modulation.
4.1 Dynamical Instabilities Due to Second Order Phase Interaction Curves.
Hansel and colleagues (Hansel and Mato, 1993; Hansel et al., 1993a, b) showed that phase-reduced models of weakly excitatory, synaptically coupled Hodgkin‘Huxley neurons elicited comparable phase-locking behaviors as long as the PIF retained at least the first two Fourier components of the original neural interaction. They employed the form,
where R is a scalar that modifies the contribution of the second order term and β de-phases the relative position of the two modes. This adjustment allows the PIF to incorporate extra biophysical detail such as specific forms of post-synaptic currents (Hansel et al., 1995). The first thing to note is that the two modes have opposite sign: whereas the first term, as in the Kuramoto model, pulls the phases toward β, the second term tends to destabilize this phase configuration. When β > 0 there exists R > R c (α) for which these two modes intersect twice and therefore the PIF vanishes at four points along a full cycle: two stable (attracting) nodes separate two unstable saddles. These extra fixed points therefore facilitate the existence of distinct clusters of phase locked oscillators even if the coupling is otherwise global, a phenomenon that is not possible with the first Fourier mode alone. As illustrated in Figure 4, if R decreases the two extra fixed points of the PIF approach each other and then annihilate in a saddle-node bifurcation as R crosses below R c .
Figure 4. (A‘C) Show the second order phase interaction employed by Hansel and colleagues with β = 0.25 and three values of R . A saddle-node bifurcation in the fixed points associated with this function occurs as the second peak in this curve crosses 0 (left to right).
This dynamic instability in the PIF enables a rich variety of more complex dynamics, most notably the emergence of heteroclinic cycles (Hansel et al., 1995). In essence, the presence of a saddle point between the cluster states engendered by the second order PIF allows for spontaneous cycling between different phase synchronous configurations. That is, in a network of N oscillators there may exist k distinct clusters each of size m 1 , m 2 ,…, m k (e. g., Tass, 1999). Through the presence of the saddle point, however, one or more of these cluster states is only marginally stable meaning that their oscillators spontaneously de-phase. These free oscillators then synchronize with other clusters, destabilizing at least one of these so that this winnerless competition continues ad infinitum . Whilst this occurs in the absence of noise, the injection of a stochastic influence into the states stabilizes the frequency of the slow heteroclinic cycling which then scales with log of the variance noise (Hansel et al., 1993a).
We note that in these systems, two distinct time scales arise naturally: the fast dynamics of the oscillators and the relatively slow rotation through the heteroclinic cycle. Systems with distinct time scales have been well studied as they often arise through mere construction, e. g., multiplication of one or more dynamical variables by an explicit time scale factor which functions to slow down the dynamics in the associated subspace (e. g., Fujimoto and Kaneko, 2003; Breakspear and Stam, 2005; Kiebel et al., 2008). By contrast, a heteroclinic cycle does not require a separation of time scales to be defined in functional form because they are an emergent property of the dynamics. The intricate sequence of cycle states and the controlled expression of instability allows a variety of putative computational functions to be enacted by such networks, even with relatively small number of oscillators. For example, Ashwin et al. (2007) showed that procession through a precisely defined cycle could function to encode a complex, sensory input as a spatiotemporal sequence in a manner that was robust to strong noise (Wordsworth and Ashwin, 2008) and could be learnt by other systems of coupled oscillators (Orosz et al., 2009).
4.2 Dynamical Instabilities and Spatial Frustration on Cortical-Like Sheets.
By combining the spatial coupling discussed in Section 3 with the second-order phase interaction function, it is possible to employ the framework of weakly coupled oscillators to understand spontaneous dynamics on cortical-like sheets, an area of strong current interest (Fox and Raichle, 2007; Honey et al., 2007; Deco et al., 2009). In particular, it is possible to explore the relationship between the dynamic instability engendered by the interaction of the two modes of the PIF and the spatial expression of frustration introduced by the phase offset. In Figure 5, we present three simulations employing the PIF defined by equation (13) and increasing the de-phasing of the PIF modes by increasing β. Following Hansel and Mato (1993), we fix the relative modulation of the PIF by the second mode to R = 0,25. Phenomena such as clustering and cycling are robust to changes in this value. In the top row (Figures 5A‘C), we plot a snapshot of the relative phases and in the bottom row (Figures 5D‘F), the local expression of the coupling influence given by equation (12). In the absence of significant de-phasing (β ≈ 0) the system evolves rapidly toward spatiotemporal patterns dominated by coherent fronts of traveling waves with small pinwheel-like patterns where these intersect (Figure 5A). Local expression of high phase frustration is apparent at these points (Figure 5D). Hence, without de-phasing between the first and second modes, the scenario is almost identical to that encountered with the spatial kernel and simple sinusoidal PIF (Figure 4). However, as β slowly increases, an instability appears at these points and grows to encompass a small patch of oscillators (Figures 5B, E). With a further increase of β these instabilities grow in spatial extent and, whilst not evident in a snapshot, begin to invade the surrounding patches of coherent wave fronts (Figures 5C, F). In consequence, the instability in the PIF, introduced by a de-phasing of the first and second Fourier modes, is inconsequential in areas of low spatial frustration. However, at points of spatial incoherence, this dynamic instability leads to a spatial instability expressed as areas of high phase disorder. This effect appears to be invariant to particular choices of the spatial kernel. In Figure 6, we illustrate an example using a spatial kernel whose outer extent is phase advancing, and which is associated with large traveling fronts organized around pinwheels. With a de-phasing of the two modes of the PIF, the centers of the pinwheels are destabilized by the same tension between phase frustration and the dynamic instability within the PIF.
Figure 5. Spatiotemporal dynamics in systems of oscillators coupled through a local spatial kernel and a second order PIF, equation (13). Columns left to right depict results for increasing phase offset β between the two modes. Top row (A‘C) shows representative oscillator states using the same color scheme as Figure 3. Bottom row (D‘F) shows the coupling tension F as defined by equation (12) where blue denotes F = 0.
Figure 6. As with Figure 5 with the exception of a larger spatial kernel whose outer extent is upgoing (phase advancing) hence engendering large coherent fronts and spiral waves.
5 Population-Level Descriptors of Cortical Rhythms.
In Section 2, we introduced the notion of the order parameter r for the Kuramoto model and showed how the governing equation could be rewritten using this quantity. In many instances, however, one is not only interested in the mean field, i. e., the mean phase ψ and its divergence r , but also in the nature of the whole probability distribution of states (for review, see Deco et al., 2008), or at the least its first few moments (mean, variance, skewness, kurtosis). This is particularly true when seeking to establish a link between cortical dynamics and cognitive processes because there is a natural mapping between the moments of neuronal states and components of cognition, including expectation, certainty, and surprise (Friston and Dolan, 2009). The distribution of states is also a crucial notion when the oscillators are influenced by stochastic forces. Finally, the probability distribution is crucial to a proper understanding of data obtained from oscillating neuronal systems because it determines the moment-to-moment statistics of these time series as the underlying system randomly samples its phase space.
In this section, we briefly overview the population formulation of the Kuramoto model, namely the nonlinear Fokker‘Planck equation, and contrast it to the linear Fokker‘Planck equation that can be derived from populations of spiking neurons. This is a crucial aspect of the Kuramoto model because it nicely recasts the circular causality that is present in the original formulation, whilst also underlining many of the important analytic results discussed in Section 2 (e. g., Strogatz and Mirollo, 1991; Acebrón et al., 2001). By knowing the distribution of states, it is also possible to estimate information-theoretic quantities such as entropy and hence provide a more direct link to notions of free energy described in Sections 3 and 4.
Before proceeding, it is crucial to underline an important distinction between the probability distribution of the states of the oscillators p on the one hand, and the population density of the whole ensemble ρ on the other. The population density is a quantitative measure of the relative states of all the oscillators and can be estimated in large purely deterministic systems as well as those that have explicit stochastic forces operating on the oscillators. The probability distribution p is the likelihood function for oscillators and only makes sense when stochastic influences have been explicitly defined. In some, quite general cases, the two are interchangeable. In the following, we first consider the population density of states in a large ensemble of Kuramoto oscillators because it follows naturally from the preceding focus on deterministic dynamics. We then introduce the stochastic Kuramoto model and consider the evolution of the probability distribution that can defined in this setting. Finally, we consider the interchangeability of p and ρ in large stochastic Kuramoto oscillators.
5.1 The Continuity Equation.
We first consider the population density formulation of the pure Kuramoto model. In the so-called thermodynamic limit of an infinite number of oscillators the mean field centroid vector, equation (4), can be written as an average over the phases and frequencies of the ensemble,
A continuity equation can be established by noting that any change in the shape of the population density, due to the drift ν n of any one or more oscillators, is governed by the Kuramoto model. Averaging over the pre-specified frequency distribution g (ω) simplifies the Kuramoto model to deviations from the mean frequency ω 0. Then the continuity equation reads.
for the sake of generality we here use a non-vanishing mean frequency ω 0.
This equation simply restates the Kuramoto model at the level of the entire population, and demands that perturbations of this function must obey the underlying deterministic equations whilst also preserving the area under the density curve. In the thermodynamic limit, significant fluctuations around the mean field vanish. However, in the finite size setting, there exists a precise, hierarchical organization of all higher order moments which scale with 1/ N , inversely with system size (Hildebrand et al., 2007). The nature and role of fluctuations in neuronal oscillations ‘ discussed further below ‘ underlines the importance of a full understanding of these moments.
5.2 Stochastic Forces and the Fokker‘Planck Equation.
All the dynamics discussed thus far incorporate stochasticity solely through the randomness of the oscillators’ frequencies ω n . Now we include stochastic forces by explicitly introducing white noise into the dynamics (3). In order to do so, we return to the finite N ensemble of phase oscillators and write the so-called Langevin equation of the stochastic Kuramoto model,
where the ξ n ( t ) are spatially independent and temporally uncorrelated random fluctuations with vanishing means and variance σ 2. These fluctuations may arise from influences that are external to the system, or they may represent a correction to the incomplete specification of a complex system as a system of weakly coupled oscillators. Naturally, in real systems, modeling such fluctuations is crucial as they inevitably occur even if only because of thermal effects. Indeed, functional forms which introduce and explicitly parameterize stochastic processes in real neuronal systems are arguably of great importance because of the mounting evidence for the functional role of noise in neurophysiological recordings (Faisal et al., 2008), perceptual performance (Moss et al., 2004) and computational models of the brain (Deco et al., 2009).
The dynamics of the joint probability of the distinct oscillators’ states, p (θ 1 ,θ 2 ,…,θ n , t ), embody a diffusion process as the stochastic fluctuations ξ n lead to divergence and hence a finite, non-zero variance in the ensemble. The diffusion supplements the deterministic, intrinsic forces caused by the interaction between the oscillators, as discussed above. The resulting mathematical form for the time-evolution of the p is referred to as the Fokker‘Planck equation of the system (Stratonovich, 1963; Risken, 1989). In fact, there is a variety of possibilities to derive the Fokker‘Planck equation for the stochastic Kuramoto model (e. g., Sakaguchi, 1988; Sakaguchi et al., 1988). Without going into detail, for (16) one finds.
where the scalar parameterizes the amplitude of the stochastic forces 5 .
As discussed earlier, the Kuramoto model can be recast using its mean field (5). In the presence of stochastic forces, this approximation becomes.
When averaging over the frequency distribution g (ω) of the natural frequencies, the dynamics reduces to.
which finally yields a simpler form of the Fokker‘Planck equation, namely.
where r is given by (14). If the variance of the noise goes to 0, then the probability density p is replaced by the population density ρ and this second-order partial differential equation reduces to the first-order continuity equation (15).
5.3 The Nonlinear Fokker‘Planck Equation.
The quantity θ does not only describe all possible states but is also a representative of all individual phases ‘ that is, phases are indistinguishable because the population has been represented as homogeneous. Put alternatively, the temporal average over states converges to the spatial average over the ensemble because the dynamics are mixing ( ergodic ). When this assumption holds, then we can readily identify the density ρ(θ, t ) of “real” oscillators θ 1 ,θ 2 ,…, discussed in Section 5.1, with the probability density p (θ, t ). That is, we use.
Substituting this into (19) yields.
here we again averaged over the frequency distribution. Following Frank et al. (2000) we write the Fokker‘Planck equation as.
Whilst this form is indeed similar to (20), by explicitly incorporated the mean field into the dynamics, it is straightforward to see that the Fokker‘Planck equation is non-linear in its probability density 6 .
The second term on the r. h.s. of (23) incorporates the tendency of the stochastic influences to scatter the density toward a uniform distribution around the unit circle. If K equals 0 then the first term vanishes and this is all that occurs. The first term embodies the deterministic tendency of the coupling amongst the oscillators to increase the density around the mean of the natural frequencies ω 0. The observation that this term is nonlinear in the density of states p is nothing other than a recasting of the circular causality we encountered in Section 2: As the system synchronizes, the density of states contracts, at the same time increasing the effective pull toward the mean. In the presence of stochastic influences, the aggregation of the members of the ensemble toward the minimum of this energy landscape, through the deterministic force is amplified by means of statistical feedback . That is, the more probable a stable state, the less it is affected by noise and, conversely, the less a stable state may be affected by noise, the more probable it is (Frank et al., 2002). This contrasts strongly with traditional accounts of diffusion under a constant force, such as regular diffusion in a harmonic potential, where the force is imposed externally and constant. Even in the presence of strong noise, it means the system can depart from classic exponential statistics, showing a tendency toward power-law scaling in the character of its temporal fluctuations (Sokolov et al., 2002; Zaslavsky, 2002). Its characterization requires novel tools from non-extensive thermodynamics like generalized entropies (Tsallis and Brigatti, 2004; Tsallis, 2006).
5.4 Contrasting Linear and Non-Linear Fokker‘Planck Equations.
It is likewise possible to derive a Fokker‘Planck representation of ensembles of spiking neurons (e. g., Deco et al., 2008), although doing so typically requires the diffusion assumption, namely that the currents arising at individual neurons are uncorrelated. This step allows higher order moments and their coupling to be discarded and leads to the drift term of the r. h.s. in (17) being linear in both the states and the density. This contrasts with the derivation of the Fokker‘Planck equation for the Kuramoto model for which correlations amongst the inputs are an indispensable property of the model, and for which, through the nonlinearity in the density, the moments of the ensemble are interdependent. The linear Fokker‘Planck equation for spiking neurons describes a process that is akin to classic diffusion in the presence of an external force and hence predicts that the distribution of the states should be approximately Normal and the spikes Poisson. The sufficient statistics in this setting include just the mean and the variance. The diffusion assumption is certainly consistent with a powerful body of research that posits a crucial role for these Gaussian statistics in the performance of optimal Bayesian inference through population coding (Ma et al., 2006; for review, see Friston and Dolan, 2009). Several recent papers have reported that the statistics of non-rhythmic activity at high frequencies (above 30 Hz) are consistent with a classic Poisson process (Bedard et al., 2006; Miller et al., 2009). Crucially, this corresponds to activity across a broad frequency range of non-rhythmic activity that has a featureless power spectral density.
The nonlinear Fokker‘Planck equation allows for a departure from these classic processes. Specifically, through an interaction between the density of the states and the effect of stochastic influences, the nonlinear Fokker‘Planck equation allows for a partially synchronized system to exhibit long dwell times near complete synchrony and associated extremal amplitude events in the mean field term, properties that have recently been documented in the beta (≈20 Hz) rhythm of human resting state EEG (Freyer et al., 2009). These non-classic statistics in the amplitude fluctuations supplement prior findings of long-tailed distributions in the temporal statistics of low frequency (below 30 Hz) rhythmic cortical activity (Linkenkaer-Hansen et al., 2001; Stam and de Bruin, 2004). Activity at lower frequencies showing non-classic statistics is associated with clear peaks in the power spectrum that are hence consistent with an underlying rhythmic process, in contrast to feature-less high frequency activity.
Another interesting departure of the nonlinear Fokker‘Planck equation from classic statistics is its ability to support the coexistence of multiple co-occurring attractor states ‘ or multistability (Frank et al., 2000). In the presence of either sufficiently large stochastic influences, or marginal attractor stability, this allows a system of coupled oscillators to erratically and spontaneously switch between different itinerantly expressed solutions. This is also of particular relevance for cortical rhythms, given the recent evidence for such bimodality in the human alpha rhythm (Freyer et al., 2009). In particular, the alpha rhythm appears to spontaneously and erratically switch between a low and high amplitude state, consistent with noise-driven switching between co-existing coherent and incoherent phase configurations amongst the underlying oscillators. It also suggests a unifying mechanism for multistability that has been observed in a number of basic cortical functions including human perception (Ditzinger and Haken, 1989), decision making (Deco and Rolls, 2006), and behavior (Schoner and Kelso, 1988).
In summary, whilst rhythmic behavior appears to conform with the anomalous statistics of the nonlinear Fokker‘Planck equation, non-rhythmic behavior appears to be consistent with uncorrelated spiking activity that conforms to the diffusion approximation. At this stage, apart from a somewhat unaesthetic partition of cortical activity into distinct correlated rhythms and uncorrelated broad frequency spiking activity, it is difficult to see how this apparent paradox can be reconciled.
6 Conclusion.
The objective of the present manuscript was to provide a neurobiologically minded overview of the essential concepts, dynamics, and analysis of the Kuramoto model, which can be considered a canonical model of synchronous oscillations in complex systems. In the purely deterministic setting, we traced the impact of introducing interaction functions and spatial embeddings that may be more representative of neuronal processes. Even in the absence of random forces, we saw that these introductions engendered a broad repertoire of rich, non-trivial dynamics. Although the framework becomes mathematically more challenging, introducing stochastic influences and studying population-wide responses is still possible. When written in this manner, we see the essential component of the Kuramoto model is preserved, namely the tendency of synchronization to become self-fulfilling, buffering the mean field against the internal noise.
We have argued that computational neuroscience can benefit from detailed physiological models at all spatial and temporal scales as well as more abstract approaches that seek deeper unifying mechanisms. This is an approach that has led to many exciting discoveries in the physical sciences that are able to unify apparently diverse phenomena. It is equally true that understanding the role of detailed mechanisms provides deeper insight into their precise mechanistic and functional roles. One challenge facing both these fields is to unify the apparently uncorrelated character of noisy spike trains, with the correlated and non-Gaussian nature of rhythmic dynamics.
We hope that the rich dynamics arising from this simple system inspires computational neuroscientists interested in the fundamental mechanisms of cortical rhythms to further investigate its historical and conceptual foundations.
Conflict of Interest Statement.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Agradecimentos.
We thank Kees Stam, Pete Ashwin, Angela Langdon, and Tjeerd Boonstra for formative discussions on the Kuramoto model and the role of oscillations in cortical systems. Michael Breakspear and Stewart Heitmann acknowledge the support of ARC Thinking Systems support and Brain Sciences UNSW. Michael Breakspear acknowledges the support of BrainNRG. Andreas Daffertshofer thanks the Netherlands Organisation for Scientific Research (NWO) for financial support.
^ The Matlab source code for our numerical simulations is available from the authors on request. ^ The sinusoidal form is “first-order” because it stems from a pair-wise linear coupling between the underlying (self-sustaining) oscillators when approximated by (almost) harmonic balance; see also section 2.3. ^ The term “order parameter” stems from statistical physics where it is used to quantify different state (or phases) often in terms of thermodynamical potentials like the free energy. In complex dynamical systems, identifying an order parameter relies on a clear-cut separation of time scales: here, r evolves significantly slower than the individual oscillators θ i . Due to this difference in time scales, all θ i can quickly adapt to changes of r , which thus prescribes the dynamics (or order), it “enslaves” the individual parts of the system (see, e. g., Haken, 1983; Tass, 1999, for more details). ^ In the study of nonlinear oscillators this averaging is also referred to as “harmonic balance” or as a combination of “rotating wave” and “slowly varying amplitude approximation.” ^ A typical strategy to solve this equation is first to look a the corresponding marginal distributions, e. g., , by which the N - dimensional state space can be iteratively reduced step-wise. Exploiting the equivalence between the different phases, then integrating (17) over all but one state space variables yields (by approximation) the dynamics of p (θ, t ) as (23) below. This procedure is closely related to a mean-field approach (see Sakaguchi, 1988; Sakaguchi et al., 1988, for more details). ^ This should not be confused with conventional Fokker‘Planck equations whose drift and diffusion coefficients depend non-linearly on the state space. These equations are linear in p , whilst (23) is not.
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Keywords: cortical oscillations, neural synchrony, Kuramoto model, Fokker‘Planck equation.
Citation: Breakspear M, Heitmann S and Daffertshofer A (2018) Generative models of cortical oscillations: neurobiological implications of the Kuramoto model. Frente. Murmurar. Neurosci. 4 :190. doi: 10.3389/fnhum.2018.00190.
Received: 05 June 2018; Accepted: 22 September 2018;
Published online: 11 November 2018.
Kai J. Miller, University of Washington, USA.
Carson Chow, University of Pittsburgh, USA; National Institutes of Health, USA.
Ole Paulsen, University of Cambridge, UK; University of Oxford, UK.
Copyright: © 2018 Breakspear, Heitmann and Daffertshofer. This is an open-access article subject to an exclusive license agreement between the authors and the Frontiers Research Foundation, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are credited.

Evolution of high-frequency systematic trading a performance-driven gradient boosting model

Affiliation: Department of Ecology and Evolution, University of Chicago, Chicago, Illinois, United States of America.
Andrea L. Graham.
Affiliation: Department of Ecology and Evolutionary Biology, Princeton University, Princeton, New Jersey, United States of America.
Bryan T. Grenfell.
Affiliation: Department of Ecology and Evolutionary Biology, Princeton University, Princeton, New Jersey, United States of America.
Nimalan Arinaminpathy.
Affiliation: Department of Infectious Disease Epidemiology, Imperial College London, London, United Kingdom.
Despite the availability of vaccines, influenza remains a major public health challenge. A key reason is the virus capacity for immune escape: ongoing evolution allows the continual circulation of seasonal influenza, while novel influenza viruses invade the human population to cause a pandemic every few decades. Current vaccines have to be updated continually to keep up to date with this antigenic change, but emerging ‘universal’ vaccines—targeting more conserved components of the influenza virus—offer the potential to act across all influenza A strains and subtypes. Influenza vaccination programmes around the world are steadily increasing in their population coverage. In future, how might intensive, routine immunization with novel vaccines compare against similar mass programmes utilizing conventional vaccines? Specifically, how might novel and conventional vaccines compare, in terms of cumulative incidence and rates of antigenic evolution of seasonal influenza? What are their potential implications for the impact of pandemic emergence? Here we present a new mathematical model, capturing both transmission dynamics and antigenic evolution of influenza in a simple framework, to explore these questions. We find that, even when matched by per-dose efficacy, universal vaccines could dampen population-level transmission over several seasons to a greater extent than conventional vaccines. Moreover, by lowering opportunities for cross-protective immunity in the population, conventional vaccines could allow the increased spread of a novel pandemic strain. Conversely, universal vaccines could mitigate both seasonal and pandemic spread. However, where it is not possible to maintain annual, intensive vaccination coverage, the duration and breadth of immunity raised by universal vaccines are critical determinants of their performance relative to conventional vaccines. In future, conventional and novel vaccines are likely to play complementary roles in vaccination strategies against influenza: in this context, our results suggest important characteristics to monitor during the clinical development of emerging vaccine technologies.
Author Summary.
Influenza vaccines used today offer good protection, but have limitations: they have to be updated regularly, to remain effective in the face of ongoing virus evolution, and they cannot be used in advance of an influenza pandemic. In this study we considered how such ‘conventional’ vaccines might compare on the population level against new ‘universal’ vaccines currently being developed, that may protect against a broad spectrum of influenza viruses. We developed a mathematical model to capture the interactions between vaccination, influenza transmission, and viral evolution. The model suggests that annual vaccination with universal vaccines could control annual influenza epidemics more efficiently than conventional vaccines. In doing so they could slow viral evolution, rather than promoting it, while maintaining the broadly protective immunity that could mitigate against the emergence of a pandemic. These effects depend sensitively on the duration of protection that universal vaccines can afford, an important quantity to monitor in their development. In future, it is likely that conventional and universal vaccines would be deployed in tandem: we suggest that they could fulfill distinct roles, with universal vaccines being prioritised for managing transmission and evolution, and conventional vaccines being focused on protecting specific risk groups.
Citation: Subramanian R, Graham AL, Grenfell BT, Arinaminpathy N (2018) Universal or Specific? A Modeling-Based Comparison of Broad-Spectrum Influenza Vaccines against Conventional, Strain-Matched Vaccines. PLoS Comput Biol 12(12): e1005204. doi:10.1371/journal. pcbi.1005204.
Editor: Roland R. Regoes, ETH Zurich, SWITZERLAND.
Received: February 16, 2018; Accepted: October 15, 2018; Published: December 15, 2018.
Copyright: © 2018 Subramanian et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: Full materials and methods are supplied in the supporting materials.
Funding: This work was supported by: Health Grand Challenges Program, Center for Health and Wellbeing, Princeton University (RS); and Bob and Cathy Solomon Undergraduate Research Fund, Princeton Environmental Institute, Princeton University (RS); and MRC Centre for Outbreak Analysis and Modelling, MR/K010174/1B (NA). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introdução.
Seasonal and pandemic influenza pose major public health challenges [1, 2] Vaccines against seasonal influenza aim to raise antibodies against the hemagglutinin (HA) and neuraminidase (NA) surface proteins of circulating strains [3]. While these targets offer the best correlates for immune protection, they are also by far the most variable amongst influenza viral components [4, 5], undergoing continual evolution for immune escape: current seasonal influenza vaccines therefore need to be updated regularly. Moreover, influenza pandemics are caused by the emergence of a virus with an altogether new HA (and other viral components), to which there is little or no immunity in the human population [6, 7]. Current vaccines cannot be deployed in advance of an influenza pandemic, as it is not possible to predict what virus will cause the next pandemic [8].
There is evidence to suggest that other viral components, including the matrix protein M1 and the nucleoprotein NP, may be more conserved than HA and NA [9–13]. Immunity to these proteins, mediated by T-cells rather than by antibodies, is associated with broad-spectrum protection[14], even against novel pandemic viruses [15, 16]. At the same time, it is also possible for antibodies to raise broad-spectrum protection: with most of HA variability concentrated in the ‘head’ region of the protein, antibodies against the more conserved (but less accessible) ‘stem’ region have also attracted considerable attention[17–19]. Antibodies against the ion channel protein M2 have also been shown to elicit broad protection [20, 21].
Only with recent advances in vaccine technology has it been possible to target these alternative viral components [10, 17–19, 22–24]. The resulting emergence of candidates for ‘universal’ vaccines raises the potential for more stable influenza vaccination programmes, that do not have to be updated so frequently. At the same time, even with current, strain-matched vaccines, population coverage is on an increasing trend: in some settings (notably in the UK) there is growing emphasis on widened vaccination coverage to reduce transmission as well as disease [25]. Coverage in the US has been steadily rising and has recently exceeded 43% of the population [26]. These trends suggest that annual, mass influenza immunization programmes could foreseeably become a reality.
Together, such developments raise important questions about the potential future use and impact of influenza vaccines. For example, how might novel vaccines compare against current, strain-matched vaccines, in their ability to control transmission? What are the implications for seasonal HA evolution, of a mass immunisation programme targeting HA versus one targeting other more conserved viral components? As these vaccines are still in development, important vaccine parameters, including classical vaccine efficacy, and duration of protection in humans, remain to be determined [27]: what are the implications of these vaccine characteristics, for future immunization programmes?
Previous work [28] focused on the emergence of a pandemic virus, finding that the ability of cross-protective vaccines to mitigate pandemic risk depended on the ability of any vaccine (whether current or future) to provide broader protection than that provided by natural infection. Another modeling study [29] showed how cross-protective vaccines could slow the rate of antigenic evolution for seasonal viruses, thus enhancing the control of seasonal epidemics with conventional (HA-specific) vaccines. However, neither model addressed the potential effect of conventional vaccines on seasonal viral evolution, and how this might compare with universal vaccines. The present model builds on this previous work, addressing the questions above with a simple, novel model of influenza transmission and evolution. The model evaluates the relative merits of ‘conventional’ versus ‘universal’ vaccination, while casting light on vaccine characteristics that would be helpful to quantify, in anticipation of novel vaccine candidates entering advanced clinical trials.
To motivate the model, we first give a brief overview of influenza vaccines. At present one of the most widely used influenza vaccines is the trivalent inactivated vaccine (TIV) [3]. Consisting only of non-replicating viral material, this formulation raises antibodies against HA and NA, but no T-cell immunity. Another formulation is the live-attenuated influenza vaccine (LAIV), using cold-adapted influenza viruses to target specific strains of HA and NA [30] Although such vaccines raise T-cell immunity through viral replication, a recent study suggests that they offer only modest efficacy against antigenically drifted strains [31] and reduced heterosubtypic immunity [32], comparable to that of TIV. Nonetheless, cross-protection could be enhanced through adjuvants or T-cell boosting [33, 34].
Meanwhile, emerging candidates for ‘universal’ vaccines focus on the exclusive expression of cross-protective immunogens, whether T-cell targets [23, 24] or the conserved HA stem [17, 19, 35]. Such vaccines have shown protection against heterosubtypic challenge in animal models [19, 24, 35], as well as in human challenge studies [36]. A limitation amongst T-cell vaccines in particular is that—unlike immunity to HA or NA—they do not block infection, but rather control the severity of disease [37]. Nonetheless, in doing so, they substantially reduce the amount and duration of viral shedding, thus reducing opportunities for transmission [24, 36].
In this context, we concentrate on the potential, future impact of mass vaccination programmes. We distinguish two types of immunity in the model: ‘strain-specific’ immunity is long-lasting and blocks infection against a given (immunizing) HA strain, and offers some protection against related strains, diminishing with antigenic distance from the immunizing strain. ‘Cross-protective’ immunity—consistent with T-cells—wanes over time [38, 39] but acts uniformly against all HA strains: conservatively, we assume that this type of immunity does not reduce susceptibility, but instead lowers infectiousness in the event of infection.
We assume that natural infection raises both types of immunity: further, we assume that ‘conventional’ vaccines (TIV and LAIV) raise only effective strain-specific immunity, and that ‘universal’ vaccines (concentrating here on T-cell vaccine candidates) raise only effective T-cell immunity. In the current work, this dichotomous choice of vaccine effect helps to contrast the corresponding population-level effects that arise. In practice, however, a ‘cocktail’ vaccine formulation could combine both types of vaccine effect: although beyond the scope of this paper, we discuss potential implications of this type of vaccine below. For a summary of the immune transitions in the host population, see Fig A in S1 Appendix.
The model has two main, coupled components: The ‘epidemic component’ captures the acquisition of immunity through vaccination and natural infection, while the ‘interepidemic component’ captures the loss of immunity in the population through antigenic drift; waning of cross-protective immunity; and population turnover. We describe these both in turn.
1. Epidemic component.
The epidemic component is a deterministic, compartmental framework that models each seasonal epidemic as a single epidemic wave, with a single circulating strain (Fig 1A). For simplicity we ignore age structure, as well as spatial heterogeneities, assuming simply a fully ‘well-mixed’ population. The governing equations are as follows:
Expand Fig 1. Overview of the two major model components.
(A) The epidemic component, a deterministic, compartmental model capturing a single influenza season. Boxes represent proportions of the population in different states, where ‘CP’ denotes cross-protective immunity. Here we assume conservatively that the role of cross-protective immunity is to reduce infectiousness without necessarily reducing susceptibility: as a result in the force-of-infection term λ , there is a coefficient c < 1 to represent diminished infectiousness amongst I (CP) . (B) The ‘interepidemic’ component, which governs the generation and selection of new strains. Given a reference, immunizing strain at d = 0, we assume that candidate viruses at increasing antigenic distance (horizontal x - axis) show increasing immune escape (blue curve). However, these candidates are also assumed to be less frequent during the interepidemic period (red curve). The selected virus is assumed simply to be that which maximises the trade-off between these factors (yellow curve). In the full model, this is calculated with respect to the different strains that individuals in the population have last been infected with, as described in the main text and in the appendix.
Here S , I , R are respectively the proportions of the population who are susceptible to infection; infectious; and recovered and immune. The superscript ( cp ) marks individuals having cross-protective immunity (but not strain-specific immunity); γ is the per-capita rate of recovery; and λ is the force of infection, given by:
Here β is the effective contact rate, multiplied by the average number of infections per infected case, and c is the reduction in infectiousness arising from cross-protective immunity, written so that c = 1 corresponds to fully transmission-blocking immunity.
The initial conditions are given by the proportion of the population that has HA-specific immunity, given prior epidemic sizes and the amount of antigenic drift that has occurred (see below), along with two types of vaccination programme, which are completed prior to each epidemic and with random coverage, irrespective of an individual’s exposure history or immune status: conventional vaccination displaces individuals from S to R and from S (cp) to R (cp) , while universal vaccination displaces individuals from S to S (cp) and R to R (cp) .
For comparability between the two types of vaccine being considered here, it is necessary to choose values for the quality of vaccine protection (vaccine ‘efficacy’) that are matched in terms of their population effect. We assume for simplicity that universal vaccination elicits the same cross-protective immunity as does natural infection, thus identifying c with the efficacy of universal vaccination. Correspondingly for conventional vaccines, we assume that efficacy derives from a proportion c of vaccinated individuals successfully acquiring strain-specific immunity (the rest remaining with their prior immune status). It is straightforward to show (see appendix) that both vaccines thus have the same effect on R 0 .
The initial conditions are given by the proportion of the population that has HA-specific immunity, given prior epidemic sizes and the amount of antigenic drift that has since occurred (see below), along with two types of vaccination programme, which are completed prior to each epidemic and with random coverage: conventional vaccination displaces individuals from S to R and from S (cp) to R (cp) , while universal vaccination displaces individuals from S to S (cp) and R to R (cp) .
2. Interepidemic component.
In the interepidemic period, we model a loss of immunity in the population due to three mechanisms: loss of strain-specific immunity through antigenic drift, loss of cross-protective immunity through waning of T-cell immunity, and a general depletion of immunity through population turnover (replacement of immune hosts by susceptible ones). These are implemented as follows.
For antigenic drift we adopt a simple deterministic framework to capture the essential role of population immunity, in driving selection for new variants (see, for example, ref [40] for a review of evidence supporting this assumption). We assume a one-dimensional axis of HA antigenic variation, a simplified representation of the distinctive ladder-like phylogeny of influenza A hemagglutinin [41]. Fig 1B shows the simple case of a single immunizing strain (a ‘reference’ strain). We denote d as the antigenic ‘distance’ between this and a candidate virus, shown on the horizontal axis. The Figure captures two essential features of antigenic evolution: first, candidates with greater d have a greater degree of immune escape and therefore a higher transmission potential [39] (blue curve). However, they arise at a lower frequency (red curve). Combining these two opposing factors to yield the ‘frequency-weighted immune escape’ (orange curve), we assume that—on a population level—the selected virus for an upcoming season is one that maximizes this quantity. Specifically, the frequency-weighted immune escape for this candidate virus is defined as:
where k is a parameter governing the relative rarity of immune escape variants. For example, in the theoretical case k = 0 there is unlimited viral diversity in the interepidemic period, thus allowing a pandemic-scale outbreak every year. At the other extreme as k → ∞, there is no generation of escape variants even in the face of population immunity: a situation similar to measles. For influenza, the scenario is intermediate. We calibrate the value of k in order to yield, at steady state, seasonal epidemics that infect roughly 10% of the population per season, consistent with the behaviour of seasonal influenza [42–44].
While eq (2) is in the simple case of a population with exposure to only one virus, over several seasons there is a series of viruses that emerge and circulate. Moreover, conventional vaccination in any given season offers protection against the virus circulating in that season, but also—to an extent diminishing with antigenic distance—against related viruses. It is thus necessary to keep track of the exposures to these viruses in the population, and to evaluate the proportion susceptible over all of these histories. Nonetheless, as we assume a one-dimensional antigenic space, it is only necessary to record the most recent infection or vaccination that individuals have undergone. Details of the necessary record-keeping are provided in the appendix.
For the waning of T-cell immunity, we assume simply that a proportion σ of individuals lose this immunity in every interepidemic period. For illustration we choose σ = 0.21, consistent with findings from early seminal work that suggested a T-cell half-life of 3 years [38]. However, it is important to note that there is considerable uncertainty around this Figure, with more recent studies suggesting that CD8 T-cell immunity can last as long as a decade, both for influenza ([45]) and for other viruses ([46]). Accordingly, we explore this uncertainty in the work below.
Table 1 shows the default parameter values used, and Fig 2 schematically summarises the procedure. Starting with a virus in a fully susceptible population, we simulate its spread using (1) (‘initiation’ in Fig 2). We then simulate the selection for a new immune escape variant using (2). Having determined this variant, we find the associated initial conditions (population susceptibility) for the subsequent epidemic season, and repeat the iteration from (1) to (2) (‘Circulation’ in Fig 2). Finally, to study how a pandemic would be affected by the conditions of immunity in this population, at year 25 we introduce a virus to which only pre-existing cross-protective immunity, and not HA-immunity, is effective (‘Pandemic’ in Fig 2).
Expand Table 1. Parameter values used in the model.
‘Default values’ are used in Fig 3, while parameter ‘ranges’ are used in Figs 4 and 5. Notes: a) For comparability between the two types of vaccines, we choose ϵ = c . b) R 0 is given by the ratio β/γ in the model: their individual values do not independently affect epidemic sizes, so it is only necessary to choose R 0 . c) Value corresponds to a mean lifetime of 70 years. d) Under the baseline values shown here, k is tuned to give seasonal epidemics infecting roughly 10% of the population. e) In the absence of a natural scale for k , we simply take half and twice the baseline value.
As described in the Methods, the simulation is initiated by a pandemic in a naïve population. In subsequent seasons we assume that strain selection happens during the interepidemic period (annotated by a virus in the Figure, and corresponding to Fig 1B). This leads to a loss of strain-specific immunity due to antigenic drift, and accompanies a loss of immunity through population turnover, as well as through decay of cross-protective immunity. We assume that routine vaccination, whether conventional or universal, occurs just prior to each seasonal epidemic (annotated by a syringe in the Figure). The epidemic that follows is governed by the eq (1) in the main text, leading to a gain of immunity in the population (corresponding to Fig 1A)). We iterate through seasons in this way, ultimately reporting the ‘seasonal epidemic size’ as the mean epidemic size between seasons 5 and 24, and the ‘pace of antigenic evolution’ as the mean distance between successive strains during this period. Finally, we simulate a pandemic in year 25, assuming a virus to which cross-protective immunity, and not strain-matched immunity, is effective.
Although showing a steady state in Fig 2, there are certain conditions where simulated seasonal influenza epidemics can show minor annual variations, as described below. Accordingly, we measure the ‘seasonal epidemic size’ as the mean epidemic size from years 5 to 24. We additionally define the ‘pandemic size’ as the size of the pandemic when introduced at year 25.
3. Sensitivity analysis.
The default parameter values shown in Table 1 (second column) are helpful for illustrating model behaviour. To examine the robustness of our model results to variation in these parameters, we then simulate the model through the range of plausible parameter values shown in Table 1 (third column). In particular, using latin hypercube sampling, we generate 10,000 parameter sets within the ranges shown. To ensure plausible epidemiology, we retain those parameter combinations yielding seasonal epidemics that infect between 5 and 20% of the population, consistent with estimates that influenza infects roughly 10% of the population each season [45–47]. Under this parameter set, we then investigate the variability in the relative performance of conventional vs universal vaccines.
Fig 3 provides a side-by-side comparison of the effects of conventional and universal vaccines, presenting three different outcomes: control of seasonal epidemics (panel A); the effect of vaccines on the pace of antigenic evolution (panel B); and the implications of seasonal vaccination for pandemic sizes (panel C). As described above, the Figure assumes equivalent vaccine efficacy and, in both cases, an annual vaccination program. The Figure is illustrative, involving only the point estimates for each of the input parameters involved (Table 1): below we examine the robustness of this qualitative behaviour under parameter variability.
Expand Fig 3. Comparative vaccine effects under illustrative parameters.
The Figure compares universal vaccines (orange) against conventional (blue), under different levels of coverage in the population. (A) The proportion of the population being infected by seasonal influenza, each year (B) The ‘pace’ of antigenic change, measured by the mean antigenic distance between successive seasons. (C) The size of a pandemic following several years of seasonal vaccination. In all Figures, cross-protective immunity is assumed to have a half-life of 3 years, and both HA-specific and universal vaccination occur annually. See Table 1 for parameter values.
First, Fig 3A illustrates how conventional and universal vaccines could have differing effects on long-term patterns of influenza transmission. While both vaccines reduce seasonal epidemic sizes, at any given level of coverage, universal vaccines appear to have a stronger effect in suppressing seasonal epidemics. Moreover, Fig 3B illustrates—consistently with previous work—that large-scale universal vaccination would slow antigenic evolution over several seasons. Notably, however, these results suggest that conventional vaccines would tend to do the opposite, potentially accelerating antigenic change. We discuss below how these effects might arise from the different types of vaccine action.
Under universal vaccination, the pace of antigenic evolution is driven to zero at sufficiently high coverage (Fig 3B, orange curve): in this regime seasonal transmission is so heavily dampened that there is little strain-specific immunity to drive selection for new variants. However, we note that seasonal epidemics—even of very small sizes—could still occur at this coverage (Fig 3A, ‘elbow’ in orange curve). This is a regime where universal vaccines interrupt transmission in the short term: over several years, however, seasonal viruses can sporadically persist, purely because of the accumulation of naïve individuals, rather than because of antigenic evolution—a situation analogous to measles ([47]) but with a substantially lower R 0 . As discussed below, however, spatial and stochastic dynamics would greatly affect these extreme cases.
Fig 3C additionally illustrates differences between the vaccines, for the size of a pandemic following several years of seasonal vaccination. The Figure illustrates that, while both types of vaccines can reduce seasonal epidemic sizes, high vaccination coverage with conventional vaccines tends to allow for increased pandemic sizes, whereas universal vaccines have the opposite effect. Moreover, pandemic sizes decline more rapidly with increasing universal vaccination coverage when there is no antigenic evolution (i. e. an increased gradient in pandemic sizes for vaccine coverage > 25%). This effect arises because interrupting transmission renders vaccination the sole source of cross-protective immunity in the population. The incremental impact of increased vaccination coverage is thus greater than in regimes allowing transmission, where infection is an additional source of cross-protective immunity.
To additionally explore the validity of these findings under parameter uncertainty, we conduct a multivariate sensitivity analysis as described in the Methods. In particular, we explore the key outputs of this analysis: the relative performance of conventional and universal vaccines, with respect to control of seasonal influenza; impact on the pace of antigenic evolution; and implications for pandemic control. Taking the first of these as an example, if g C is the average seasonal epidemic size under a given vaccination coverage, and g U is the corresponding quantity for a universal vaccine, we calculate the ratio r = g U / g C . As long as this quantity is below 1, the qualitative finding in Fig 3A holds true. Defining r as the ‘relative efficiency’ in control of seasonal epidemics, we likewise consider relative efficiencies in controlling antigenic evolution, and in pandemic control (corresponding to each of the panels in Fig 3).
Fig 4 plots these relative efficiencies, together with their uncertainty, for different levels of vaccination coverage. In each panel, the region above the dashed line (i. e. a ratio > 1) corresponds to conventional vaccines being more efficient than universal vaccines, and vice versa. Fig 4B and 4C suggest that universal vaccines are robustly more efficient in controlling antigenic evolution and in mitigating pandemic sizes. Notably, however, the uncertainty bounds in Fig 4A straddle the line r = 1 (shown ‘dashed’), indicating certain parameter combinations under which a universal vaccine could allow greater seasonal epidemics than a conventional vaccine.
Expand Fig 4. Relative performance of conventional and universal vaccines, under different levels of vaccination coverage.
Shaded areas represent the range of outcomes arising from the parameter ranges shown in Table 1: lower and upper boundaries depict the 2.5 th and 97.5 th percentiles of simulated outcomes, respectively. In each panel. where ratios are greater than 1, conventional vaccines have greater efficiency than universal vaccines, and vice versa. (A) Relative performance in controlling seasonal epidemics, defined as the ratio of seasonal epidemic sizes under a given coverage of vaccination, comparing conventional and universal vaccines (i. e. upper and lower curves in Fig 3A). (B) Relative performance in controlling antigenic evolution, defined as the ratio in the pace of antigenic evolution, comparing conventional with universal vaccines (i. e. upper and lower curves in Fig 3B). (C) Relative performance in controlling pandemics, defined as the ratio of pandemic sizes, comparing conventional with universal vaccines (i. e. upper and lower curves in Fig 3C).
To identify which parameters are driving this result, taking a vaccination coverage of 15%, Fig 5 shows scatter plots of the relative efficiency r with respect to each of the parameters in the model. Points of interest ( r > 1) are shown in red, and are roughly evenly distributed for each of the parameters, with the notable exception of h (fourth panel), where values of r > 1 clearly cluster around a low duration of protection. Motivated by this Figure, holding h constant at its default value, and re-sampling other parameters, yields values of r strictly less than 1 (see Fig C in S1 Appendix). Overall, therefore, in the range of parameter values explored here, universal vaccines appear robustly more efficient in controlling seasonal epidemics, as long as the duration of protection that they provide is sufficiently long.
Expand Fig 5. Sensitivity analysis of relative efficiency in controlling seasonal epidemics.
Relates to Fig 4A, and assuming 15% coverage. The quantity r is the relative seasonal epidemic size under 15% coverage of a universal vaccine, relative to that under the same coverage of a conventional vaccine. Points with r > 1 (shown in red) universal vaccines are less efficient than conventional vaccines, and vice versa. Each panel shows results over 10,000 simulations spanning the parameter ranges in Table 1 (third column), as a scatter plot with respect to each of the parameters in the model. Note here that each point has the same height in each panel; they are simply arranged in different ways along the horizontal axis, depending on their relationship to the model parameter denoted on that axis. The median and 95% intervals for r are 0.38 (0.03–0.92). Parameters are as follows: R 0 , basic reproduction number; c , effect of cross-protective immunity in reducing transmission potential; k , abundance of immune escape variants, relative to immune escape potential; h , half-life of cross-protective immunity; 1/ μ , mean host lifetime.
Discussão.
While a major focus in the development of new influenza vaccines is on their ability to provide individual protection, anticipating the population-level effects of vaccination can also yield useful public health insights. Here, we present a simple model bringing together influenza evolution and epidemiology, and use this model to compare vaccination programmes with two different types of influenza vaccine: current, ‘conventional’ strain-matched vaccines, versus emerging, ‘universal’ vaccines.
A primary result from this work is the contrasting evolutionary effects associated with the two types of vaccines. In general, sustained control of transmission reduces the number of immune individuals in the population, and thus dampens selection pressure for new antigenic variants (Fig 3B): universal vaccines, because they are not HA-specific, are able to achieve this state without themselves contributing to HA selection pressure. Thus the pace of antigenic evolution decreases with higher universal vaccine coverage. However, conventional vaccines raise strain-matched immunity: they therefore have the opposite effect to universal vaccines, compounding HA selection pressure and thus tending to accelerate antigenic evolution.
In control of seasonal influenza, universal vaccines could also avert more transmission per dose administered than conventional vaccines, with the potential to interrupt transmission even at moderate levels of coverage (Fig 3A). This amplified effect likely arises from the fact that universal vaccination reduces strain-matched immunity in the population while increasing cross-protective immunity and slowing antigenic drift, while conventional HA-specific vaccines do the converse. Overall, therefore, universal vaccination could complement population immunity in a way that is more efficient for controlling transmission, over several seasons, than strain-matched vaccines. Furthermore, while the effects of HA-specific vaccination are limited by antigenic evolution, the effects of universal vaccination are limited by the duration of cross-protective immunity [28]. As long as this duration is long enough to persist across vaccination intervals, the effect of cross-protective vaccination can be maintained on a population level with each passing season.
Conventional vaccines provide HA-specific immunity against seasonally circulating strains at the expense of infection-acquired immunity that may otherwise protect against novel antigenic subtypes [48–50]. Thus, as shown in Fig 3C, increased HA-specific vaccination coverage could result in increased pandemic sizes. Indeed, these model findings are consistent with experimental findings in animal challenge studies [50]. By contrast, a universal vaccine, even if transmission-blocking rather than infection-blocking, could reduce pandemic sizes by promoting cross-protective immunity in the population. Similar phenomena have been suggested by Zhang et al via different mechanisms, by which cross-protective immunity limits the opportunities for reassortment, thus limiting the emergence of pandemic-capable viruses ([28]). Taken together, these findings suggest that universal vaccines could be effective in both preventing and mitigating pandemic emergence.
While influenza is a readily evolving pathogen, it is evidently not so rapidly evolving as to cause pandemic-scale epidemics every season. Here, we capture this phenomenon by assuming that viral evolution is limited by the available HA diversity in the population (Fig 1B). As for the conserved antigens targeted by universal vaccines, we have ignored the potential for immune escape, assuming in this work that any antigenic change would be too functionally costly for the virus to continue replicating. Nonetheless, the potential for such immune escape cannot be discounted: should it occur it would have far reaching consequences, comparable to pandemic emergence. Additionally, even if conventional vaccines should have negative implications for pandemic control, for their sterilizing immunity they would remain essential in routine immunization to protect specific risk groups such as the immunocompromised and the elderly.
Overall then, rather than replacing one vaccination programme with another, it is important to consider universal vaccines as being strategically complementary to conventional, strain-matched vaccines. With recent work highlighting the potential effects of influenza vaccination programs in controlling transmission [25], our work suggests that—depending on the characteristics of new vaccines including duration of protection and vaccination frequency—the ‘transmission dampening’ role could be one best filled by universal vaccines. An alternative could be a ‘cocktail’ formulation consisting of a combination of strain-specific and vectored, cross-protective immunogens. Such cocktails could continue to protect clinical risk groups such as the elderly, as well as maintaining cross-protective immunity in the population to mitigate pandemic risk. However, their effect on seasonal influenza evolution would depend on the relative strengths of strain-specific and cross-protective protection that they provide: future work could explore the extent to which the cross-protective component of a cocktail vaccine could mitigate the potential ‘evolution-speeding’ effects of its strain-specific component.
The present model has several limitations to note. First, it involves a stylized model of influenza evolution: in practice, the antigenic dynamics of influenza arise from a combination of complex processes, spanning the chance emergence of an immune escape variant in an infected host; the transmission of that mutant to other hosts; and its successful establishment in the global population, all in competition with other potential escape variants ([51, 52]). Each of these stages is stochastic, giving rise to notable irregularities in influenza evolution such as antigenic ‘jumps’ shown by influenza A, every 3–9 years, with important consequences for vaccine selection [41] There is also notable variation in the geographical source of circulating influenza strains each year. [53] Nonetheless, the aim of the present work is not to explain such spatiotemporal variation, but rather to capture the essential, long-term interplay between population immunity and viral evolution. Consequently our current findings for universal vaccines (particularly, that they could slow antigenic evolution) are consistent with previous work, which employed a more complex, stochastic framework [29, 54]: we would expect our current findings for conventional vaccines to be similarly qualitatively robust to stochasticity in viral evolution.
Second, the model does not take into account heterogeneities such as age structure [45,46]. With school-age children playing an important role in the transmission of influenza ([55–57]), and the elderly being less important for transmission, the effect of a given population coverage of vaccination will depend critically on how it is distributed amongst age groups ([58]). Neglecting such effects, our model may overestimate the impact of a given vaccination coverage, for example, suggesting such low seasonal epidemic sizes at current levels of coverage in the US (Fig 3A). If this bias applies equally for universal as well as for conventional vaccines, it may not be expected to influence our overall results about their relative efficiencies. Moreover, an important area for future work would be the potential impact of age-targeted vaccination programmes for emerging, transmission-controlling vaccines.
Several important caveats about immunity also bear mention: first, in the absence of relevant data, we have assumed that vaccine-induced immunity has an efficacy equivalent to its counterpart in natural immunity. Further work could explore the implications of relaxing this assumption. It might be expected that model results would depend to a large extent on whether vaccine-induced immunity would be more or less effective (or long-lasting) than its counterpart in natural immunity. (It is notable, for example, that recombinant technology raises the prospect of focusing immunity on particular antigens to a greater extent than is possible through natural immunity [22, 59]). Second, for simplicity we have neglected the potential for complex interactions such as between antibody-mediated and T-cell mediated immunity, and the potential effect of an individual’s infection history on their vaccine response [60–64]. These complexities are only starting to be explored for influenza, and in future a better understanding of these immunological interactions will allow refined models to explore their implications. Third, we have assumed that universal vaccination does not protect against infection (i. e. no reduction in susceptibility). This being a conservative assumption, we might expect our overall findings to be accentuated by allowing for such additional protection. Conversely, we have assumed that current, strain-matched vaccines elicit no cross-protective immunity. Although this is a helpful caricature for contrasting two different modes of vaccination, conventional vaccines may also elicit some heterosubtypic immunity [65]. In practice, any such protection is unfortunately too weak for current vaccines to protect against novel pandemic strains [32], a major rationale for universal vaccines [23, 35]–nonetheless, any broad protection from current vaccines would tend to narrow the gap between conventional and universal vaccines illustrated in Fig 3. Such caveats notwithstanding, our overall findings are likely to hold true: a vaccine formulation enhancing cross-protective over strain-specific immunity would have qualitatively different population implications from one that does the converse. Overall, a key data need in future is a quantitative comparison of the duration and potency of cross-protection raised by current vaccines, against that offered by emerging vaccine candidates.
In summary, emerging vaccine technology, along with increasing interest in understanding the biology of influenza evolution, are offering fresh prospects for the control of influenza. In the context of these and other developments, it is becoming increasingly important to understand the role of the various arms of natural and vaccine-induced immunity in controlling influenza, and in driving viral evolution. By aiming to link these critical host mechanisms to important phenomena on the population level, mathematical models, such as the one presented here, can be valuable in casting light on the potential impact of new and emerging vaccines.
Supporting Information.
Author Contributions.
Conceived and designed the experiments: NA ALG BTG. Performed the experiments: RS. Analyzed the data: RS ALG NA BTG. Wrote the paper: RS NA ALG BTG.
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