Analysis of Synchronization in a Neural Population by a Population Density Approach

Abstract

Abstract. In this paper we deal with a model describing the evolution in time of the density of a neural population in a state space, where the state is given by Izhikevich's two -dimensional single neuron model. The main goal is to mathematically describe the occurrence of a significant phenomenon observed in neurons populations, the synchronization. To this end, we are making the transition to phase density population, and use Malkin theorem to calculate the phase deviations of a weakly coupled population model.

A Density Model for a Population of Theta Neurons

“…A well-posedness result for a population density model of Leaky Integrate-and-Fire (LIF) neurons can be found in [11]. The approach proved to be an useful tool in analyzing special behaviors of neural populations, such as the existence of equilibrium solution ( [12]), or the emergence of synchronization of neurons ( [13][14][15]). …”

Population density models of integrate-and-fire neurons with jumps: well-posedness

“…Today this approach has been enlarged: using more realistic models with two or three state variables describing the dynamics of a single neuron, one can derive a two or three dimensional PDE, see for instance [4], [15] and [9]. Such PDEs are obviously hard to simulate and hard to analyse.…”

“…Such PDEs are obviously hard to simulate and hard to analyse. A lot of deep mathematical tools were used to moderate the computational time such as moment reduction [15], phase reduction [9] and [7]. In this paper we will study only one dimensional models, namely integrate-and-fire models where the only state variable is the potential of the neuron.…”

“…If the first goal of the population density approach was to reduce the computational time to simulate a huge population of neurons, the interest of the PDE approach seems today more related to understanding the behavior of a coupled population of neurons. As we know, see [10], [7], [8], [21], [22] and [9], coupled populations of neurons can exhibit various behaviours. For example oscillations of the activity of neuronal populations at various frequencies can be observed in different brain areas in vivo in normal as well as pathological condition.…”

Quantitative Neuroscience: From Chalk Board to Bedside

“…Obviously the most promising mechanisms for neural processing are those which are robust in the presence of noise and delay. In other words, it is not only necessary to develop analytical expressions that describe neural synchronization regardless of the number of neurons [5], it is also necessary to understand the effects of noise of the dynamics of synchronizing neural populations [2]. Similarly the widespread occurrence of time-delayed feedback in neural pathways raises questions as to the role of time delays in information processing [10] and whether new effects arise from the interplay between noise…”