Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states

AMS Subject Classification: 35K60, 82C31, 92B20.The classical description of the dynamics of a large set of neurons is based on deterministic/stochastic differential systems for the excitatory-inhibitory neuron network [1,2]. One of the most classical models is the so-called noisy leaky integrate and fire (NLIF) model. Here, the dynamical behavior of the ensemble of neurons is encoded in a stochastic differential equation for the evolution in time of membrane potential v ( t ) of a typical neuron representative of the network. The neurons relax towards their resting potential V L in the absence of any interaction. All the interactions of the neuron with the network are modelled by an incoming synaptic current I ( t ) . More precisely, the evolution of the membrane potential follows, see [3-8]where C m is the capacitance of the membrane and g L is the leak conductance, normally taken to be constants with τ m = g L / C m ≈ 2 ms being the typical relaxation time of the potential towards the leak reversal (resting) potential V L ≈ ？ 70 mV . Here, the synaptic current takes the form of a stochastic process given by:where δ is the Dirac Delta at 0. Here, J E and J I are the strength of the synapses, C E and C I are the total number of presynaptic neurons and t E j i and t I j i are the times of the j th -spike coming from the i th -presynaptic neuron for excitatory and inhibitory neurons respectively. The stochastic character is embedded in the distribution of the spike times of neurons. Actually, each neuron is assumed to spike according to a stationary Poisson process with constant probability of emitting a spike per unit time ν. Moreover, all these processes are assumed to be independent between neurons. With these assumptions the average value of the current and its variance are given by μ C = b ν with b = C E J E ？ C I J I and σ C 2 = ( C E J E 2 + C I J I 2 ) ν . We will say that the network is average-ex