Spontaneous dynamics of synaptic weights in stochastic models with pair-based spike-timing-dependent plasticity

We investigate spike-timing dependent plasticity (STPD) in the case of a synapse connecting two neuronal cells. We develop a theoretical analysis of several STDP rules using Markovian theory. In this context there are two different timescales, fast neuronal activity and slower synaptic weight updates. Exploiting this timescale separation, we derive the long-time limits of a single synaptic weight subject to STDP. We show that the pairing model of presynaptic and postsynaptic spikes controls the synaptic weight dynamics for small external input on an excitatory synapse. This result implies in particular that mean-field analysis of plasticity may miss some important properties of STDP. Anti-Hebbian STDP favors the emergence of a stable synaptic weight. In the case of an inhibitory synapse the pairing schemes matter less, and we observe convergence of the synaptic weight to a nonnull value only for Hebbian STDP. We extensively study different asymptotic regimes for STDP rules, raising interesting questions for future work on adaptative neuronal networks and, more generally, on adaptative systems.

Figure 1

Synaptic plasticity kernels for pair-based rules. Example of an exponential plasticity curve as a function of $\mathrm{\Delta }t$, with different parameters (top left). Examples of pre- and postsynaptic pairings for all-to-all (PA, bottom left), nearest-neighbor symmetric (PNS, top right), and nearest-neighbor symmetric reduced (PNR, bottom right) STDPs.

Figure 2

All-to-all pair-based STDP for an excitatory synapse. (a) Dynamics of the synaptic weight as a function of $B_1$ and $B_2$ for $\nu =0$. (left) Classification based on numerical simulations for different asymptotic dynamics of the synaptic weights (see different colors). Theoretical boundaries are also drawn (see legend for their expressions). (right) Two examples of temporal evolutions of the synaptic weights, for different values of $\varepsilon$. (b) Same as (a) for $\nu =1$ (${\gamma }_1={\gamma }_2=\beta =\lambda =1$).

Figure 3

Different pairing schemes leads to diverse dynamics. (a) Dynamics of the synaptic weight as a function of $B_1$ and $B_2$ for nearest-neighbor symmetric STDP and $\nu =0$. (left) Classification based on numerical simulations for different asymptotic dynamics of the synaptic weights (see different colors). Theoretical boundaries are also drawn (see legend for their expressions). (right) Two examples of temporal evolutions of the synaptic weights, for different values of $\varepsilon$. (b) Same as (a) for nearest-neighbor symmetric reduced STDP and $\nu =0$. (c) Same as (a) for nearest-neighbor symmetric reduced STDP and $\nu =0.2$ (${\gamma }_1={\gamma }_2=\beta =\lambda =1$).

Figure 4

Pair-based STDP for an inhibitory synapse. (a) Dynamics of the synaptic weight as a function of $B_1$ and $B_2$ for all-to-all STDP and $\nu =1$. (left) Classification based on numerical simulations for different asymptotic dynamics of the synaptic weights (see different colors). Theoretical boundaries are also drawn (see legend for their expressions). (right) Two examples of temporal evolutions of the synaptic weights, for different values of $\varepsilon$. (b) Same as (a) for nearest-neighbor symmetric STDP and $\nu =1$. (c) Same as (a) for nearest-neighbor symmetric reduced STDP and $\nu =1$ (${\gamma }_1={\gamma }_2=\beta =\lambda =1$).