Integrate and fire neural networks, piecewise contractive maps and limit cycles

Abstract

We study the global dynamics of integrate and fire neural networks composed of an arbitrary number of identical neurons interacting by inhibition and excitation. We prove that if the interactions are strong enough, then the support of the stable asymptotic dynamics consists of limit cycles. We also find sufficient conditions for the synchronization of networks containing excitatory neurons. The proofs are based on the analysis of the equivalent dynamics of a piecewise continuous Poincaré map associated to the system. We show that for efficient interactions the Poincaré map is piecewise contractive. Using this contraction property, we prove that there exist a countable number of limit cycles attracting all the orbits dropping into the stable subset of the phase space. This result applies not only to the Poincaré map under study, but also to a wide class of general n-dimensional piecewise contractive maps.

Particle systems with a singular mean-field self-excitation. Application to neuronal networks

“…Along the cascade time axis at time t, neurons in Γ k+1 (k + 1 < ℓ) spike after neurons in Γ 0 ∪ · · · ∪ Γ k . We can then define Γ := 0 k N −1 Γ k , which is exactly the set of all neurons that spike at time t, according the natural ordering of the spike cascade (see also [6]). Having determined this, it is then straightforward to perform the final update of all the neurons in the network by setting…”

Locally Contractive Dynamics in Generalized Integrate-and-Fire Neurons

“…Piecewise contraction analysis is the natural extension of contraction analysis to dynamical systems that compute symbolically, as neural systems appear to do. It has been shown that under certain conditions, spiking neural networks are piecewise contractive (Catsigeras and Guiraud, to appear; Cessac, 2008). A limitation of these works is that their predictions are not experimentally verifiable.…”

“…Piecewise contractive maps Φ : V → V , where V is a metric space, satisfy the contractive property d ( x, y ) > d (Φ( x ), Φ( y )) but only when there is an i such that x, y ∈ S i , where ${false\{S_{i}false\}}_{i=1}^n$ is a partitioning of the domain of Φ such that V = ∪ S i and S i ∩ S j = ∅ for i ≠ j . Piecewise contractions have recently been used to study spiking neural networks (Catsigeras and Guiraud, to appear; Cessac, 2008). Piecewise contractive maps generate symbolic sequences by traversing the various partitions of the domain, yet they generally avoid chaotic regimes.…”

Dynamics of large cooperative pulsed-coupled networks

“…The main differences between the dynamical system that we study here and the one studied in [4], are the following: First, we assume that the cells are cooperative (which in Neuroscience are called excitatory). This means that ∆ i,j ≥ 0 for all ordered pairs (i, j) such that i = j, and the value zero is admitted in some of our results.…”

“…This means that ∆ i,j ≥ 0 for all ordered pairs (i, j) such that i = j, and the value zero is admitted in some of our results. In [4], any sign of ∆ i,j is admitted by hypothesis, but only nonzero values are assumed. Second, we do not assume that the free dynamics is the same for all the cells.…”