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Doctoral Thesis
DOI
10.11606/T.45.2016.tde-01062016-162917
Document
Author
Full name
Guilherme Ost de Aguiar
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2015
Supervisor
Committee
Galves, Jefferson Antonio (President)
Cassandro, Marzio
Ferrari, Pablo Augusto
Locherbach, Eva
Maus, Jacob Ricardo Fraiman
Title in Portuguese
Limite hidrodinâmico para neurônios interagentes estruturados espacialmente
Keywords in Portuguese
Limite hidrodinâmico
Processos markovianos determinísticos por partes
Sistemas de partículas interagentes
Sistemas neuronais
Abstract in Portuguese
Nessa tese, estudamos o limite hidrodinâmico de um sistema estocástico de neurônios cujas interações são dadas por potenciais de Kac que imitam sinapses elétricas e químicas, e as correntes de vazamento. Esse sistema consiste de $\ep^$ neurônios imersos em $[0,1)^2$, cada um disparando aleatoriamente de acordo com um processo pontual com taxa que depende tanto do seu potential de membrana como da posição. Quando o neurônio $i$ dispara, seu potential de membrana é resetado para $0$, enquanto que o potencial de membrana do neurônio $j$ é aumentado por um valor positivo $\ep^2 a(i,j)$, se $i$ influencia $j$. Além disso, entre disparos consecutivos, o sistema segue uma movimento determinístico devido às sinapses elétricas e às correntes de vazamento. As sinapses elétricas estão envolvidas na sincronização do potencial de membrana dos neurônios, enquanto que as correntes de vazamento inibem a atividade de todos os neurônios, atraindo simultaneamente todos os potenciais de membrana para $0$. No principal resultado dessa tese, mostramos que a distribuição empírica dos potenciais de membrana converge, quando o parâmetro $\ep$ tende à 0 , para uma densidade de probabilidade $ho_t(u,r)$ que satisfaz uma equação diferencial parcial nâo linear do tipo hiperbólica .
Title in English
Hydrodynamic limit for spatially structured interacting neurons
Keywords in English
Hydrodynamic limit
Interacting particle systems
Neuronal systems
Piecewise deterministic Markov process
Abstract in English
We study the hydrodynamic limit of a stochastic system of neurons whose interactions are given by Kac Potentials that mimic chemical and electrical synapses and leak currents. The system consists of $\ep^$ neurons embedded in $[0,1)^2$, each spiking randomly according to a point process with rate depending on both its membrane potential and position. When neuron $i$ spikes, its membrane potential is reset to $0$ while the membrane potential of $j$ is increased by a positive value $\ep^2 a(i,j)$, if $i$ influences $j$. Furthermore, between consecutive spikes, the system follows a deterministic motion due both to electrical synapses and leak currents. The electrical synapses are involved in the synchronization of the membrane potentials of the neurons, while the leak currents inhibit the activity of all neurons, attracting simultaneously their membrane potentials to 0. We show that the empirical distribution of the membrane potentials converges, as $\ep$ vanishes, to a probability density $ho_t(u,r)$ which is proved to obey a nonlinear PDE of Hyperbolic type.
 
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TESE_GUILHERME.pdf (641.78 Kbytes)
Publishing Date
2016-07-26
 
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