A Quasi-Stationary Approach to Metastability in a System of Spiking Neurons with Synaptic Plasticity

TL;DR

It is argued that metastable states constitute candidates for the type of transient information storage required by working memory and for the existence and uniqueness of a quasi-stationary distribution (QSD) by considering the relative orders of magnitude of the relaxation and absorption times.

Abstract

After reviewing the behavioral studies of working memory and of the cellular substrate of the latter, we argue that metastable states constitute candidates for the type of transient information storage required by working memory. We then present a simple neural network model made of stochastic units whose synapses exhibit short-term facilitation. The Markov process dynamics of this model was specifically designed to be analytically tractable, simple to simulate numerically and to exhibit a quasi-stationary distribution (QSD). Since the state space is finite this QSD is also a Yaglom limit, which allows us to bridge the gap between quasi-stationarity and metastability by considering the relative orders of magnitude of the relaxation and absorption times. We present first analytical results: characterization of the absorbing region of the Markov process, irreducibility outside this absorbing region and consequently existence and uniqueness of a QSD. We then apply Perron-Frobenius spectral analysis to obtain any specific QSD, and design an approximate method for the first moments of this QSD when the exact method is intractable. Finally we use these methods to study the relaxation time toward the QSD and establish numerically the memorylessness of the time of extinction.

Figures (13)

• Figure 1: An illustration where $\theta = 5$. Each node represents a possible membrane potential value, the first element of the pair (being understood that "5" should be interpreted as "$\ge 5$") and a synaptic facilitation value in the second element of the pair. If the neuron has it synapse facilitated it sits in the outer circle (light blue circle), if its synapse is not facilitated it sits in the inner circle (light orange squares). The transition rates are encoded by the color: brown is the rate of "effective spikes" generated by the network (spike occurring while the synapse of the spiking neuron is still facilitated), dark green is $\lambda$ and red is $\beta$.
• Figure 2: Trajectories between time units 1 and 2 of the membrane potentials of the fifty neurons of a simulated network. The traces are blue when the synapse of the neuron is facilitated and orange otherwise.
• Figure 3: Enlarge display between times 1.20 and 1.25 of the data shown of Fig. \ref{['fig:MPP1']}.
• Figure 4: Raster plots of a $50$ neurons network, with $\lambda = 6.7$, $\beta = 10$ and $\theta = 5$. Left, from time 0 to 14; right from time 12 to 14. Dots are blue when the synapse is facilitated and orange otherwise
• Figure 5: Same as Fig. \ref{['fig:raster-simA']} but different random numbers sequence. The scale bar is drawn between time 10 and time 15.