Infinite Systems of Interacting Chains with Memory of Variable Length—A Stochastic Model for Biological Neural Nets

Galves, Antonio; Löcherbach, Eva

doi:10.1007/s10955-013-0733-9

Cited by 123 publications

(184 citation statements)

References 26 publications

(28 reference statements)

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“…This is a special case of the general GL model21, with the filter function , where t s is the time of the last firing of neuron i . We have X i [ t + 1] = 1 with probability Φ( V i [ t ]), which is called the firing function 213839404142.…”

Section: The Modelmentioning

confidence: 99%

“…We have X i [ t + 1] = 1 with probability Φ( V i [ t ]), which is called the firing function 213839404142. We also have X i [ t + 1] = 0 if X i [ t ] = 1 (refractory period).…”

Section: The Modelmentioning

confidence: 99%

“…Alternatively, the Galves-Löcherbach (GL) model2138394041 and also the model used by Larremore et al 4243. introduce stochasticity in their firing neuron models in a different way.…”

mentioning

confidence: 99%

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Phase transitions and self-organized criticality in networks of stochastic spiking neurons

Brochini

Costa

Abadi

et al. 2016

Sci Rep

102

139

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Phase transitions and critical behavior are crucial issues both in theoretical and experimental neuroscience. We report analytic and computational results about phase transitions and self-organized criticality (SOC) in networks with general stochastic neurons. The stochastic neuron has a firing probability given by a smooth monotonic function Φ(V) of the membrane potential V, rather than a sharp firing threshold. We find that such networks can operate in several dynamic regimes (phases) depending on the average synaptic weight and the shape of the firing function Φ. In particular, we encounter both continuous and discontinuous phase transitions to absorbing states. At the continuous transition critical boundary, neuronal avalanches occur whose distributions of size and duration are given by power laws, as observed in biological neural networks. We also propose and test a new mechanism to produce SOC: the use of dynamic neuronal gains – a form of short-term plasticity probably located at the axon initial segment (AIS) – instead of depressing synapses at the dendrites (as previously studied in the literature). The new self-organization mechanism produces a slightly supercritical state, that we called SOSC, in accord to some intuitions of Alan Turing.

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Section: The Modelmentioning

confidence: 99%

“…We have X i [ t + 1] = 1 with probability Φ( V i [ t ]), which is called the firing function 213839404142. We also have X i [ t + 1] = 0 if X i [ t ] = 1 (refractory period).…”

Section: The Modelmentioning

confidence: 99%

See 1 more Smart Citation

Phase transitions and self-organized criticality in networks of stochastic spiking neurons

Brochini

Costa

Abadi

et al. 2016

Sci Rep

102

139

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“…Although simple, the class of models studied in this chapter is a starting point for readers who are not familiar with Galves and Löcherbah [GL13] and De Masi et al [DMGLP15] models. Moreover, this class helps to understand the intuition behind some proofs of the results presented in the Chapter 3.…”

Section: Chapter 2 Jumping Process With Memory Of Variable Lengthmentioning

confidence: 99%

“…A more interesting approach of integrate-and-fire model was proposed by Galves and Löcherbach in [GL13]. They introduce a discrete-time class of models which is a non Markovian system of infinite interacting chains with memory of variable length.…”

Section: Introductionmentioning

confidence: 99%

Stochastic models in neurobiology: from a multiunitary regime to EEG data

Oliveira¹

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Durante o desenvolvimento deste trabalho a autora recebeu auxlio financeiro da CAPES e CNPq. AbstractIn this thesis we study three different stochastic processes describing the brain activity. The first one is a continuous time version of the stochastic chains with memory of variable length.These stochastic chains take values in the set of neurons and assign, at time t, the value of the last neuron which spiked up to time t. Moreover, we assume neurons interact through a phenomena called chemical synapses. Briefly this means that when a neuron spikes, it loses all its membrane potential and at same time changes the membrane potential of the neurons which are influenced by it. Under this approach we proved the positive recurrent of the process and presented a perfect simulation algorithm able to generate a finite sample of the process under its invariant measure.In the second model we continue considering the chemical synapses interaction and add also an interaction through electrical synapses. The last one happens duo to the presence of specific channels which allow the passage of ions along the the membrane of two neurons and, as consequence, we have a sharing of potential between the neurons. Moreover, we consider also the constant lost of potential of the neurons for the environment which push each neuron to a resting state. For this model we study the long-run behaviour of the process with a finite number of neurons, the hydrodynamic limit for the system and investigate the possible invariant distributions for the limiting process.In the last model considered here we study the brain activity measured through EEG data. We investigate the predictive coding principle which says that neural networks are able to learn the statistical regularities inherent in a stimuli and reduce redundancy by removing the predictable components of the input. To test this conjecture we propose procedures to perform statistical model selection on the EEG data in order to retrieve structural features of stochastic sources. This is done through a case study in which the EEG data is recorded under the effect of two different stochastic rhythmic sources produced by two different context tree models. We present a suitable class of stochastic processes, called here as hidden context tree models, to model EEG signals evoked by rhythmic structures. Then, we propose a consistent statistical procedure to perform statistical model selection in this class and in our case study.Key words : chains with memory of variable length, piecewise deterministic Markov process, limiting distribution, neuronal systems, hidden context tree models, statistical models selection ResumoNessa tese estudamos três diferentes processos estocásticos descrevendo a atividade cerebral. O primeiro processoé uma versão a tempo contínuo das cadeias estocásticas com memória de alcance variável. Essas cadeias tomam valores no conjunto dos neurônios e assumem, no instante t, o valor doúltimo neurônio a disparar antes de t. Além disso, assumimos que os neurônios interagem en...

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Regularity of the Invariant Measure and Non‐parametric Estimation of the Jump Rate

Hodara¹,

Krell²,

Löcherbach³

2018

Statistical Inference for Piecewise‐deterministic Markov Processes

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Infinite Systems of Interacting Chains with Memory of Variable Length—A Stochastic Model for Biological Neural Nets

Cited by 123 publications

References 26 publications

Phase transitions and self-organized criticality in networks of stochastic spiking neurons

Phase transitions and self-organized criticality in networks of stochastic spiking neurons

Stochastic models in neurobiology: from a multiunitary regime to EEG data

Regularity of the Invariant Measure and Non‐parametric Estimation of the Jump Rate

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