Network burst generation by short-term plasticity

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Introduction

This example reproduces the results of (Tsodyks, Uziel, & Markram 2000; personal communication with M Tsodyks 2006). The script tsodyks_shortterm.sli sets up a network of 500 (400 excitatory, 100 inhibitory) neurons randomly connected by short-term facilitating and depressing synapses. The script tsodyks_shortterm.m creates 3 figures similar to the ones in the original publication showing a raster diagram of the spiking activity in the network (Fig.1), a histogram of population activity (Fig.2), and the distribution of spike rates (Fig.3). The prominent feature of this network is the occurence of synchronous network bursts interspaced with long periods of irregular activity at low rates.

The model described here differs from the model used in the original publication as follows:

• The subthreshold dynamics of the neurons and the short term plasticity of the synapses are integrated exactly using the method described in (Rotter & Diesmann 1999).
• The delay of the synaptic connections is 0.25 ms. The original model does not incorporate delays.
• The original model draws the values for the time constants of facilitation and recovery from the distributions specified below. The implementation described here uses the mean of these distribution as the parameter for all synapses.

These modifications do not qualitatively affect the results.

Fig. 1 Spiking activity of the network. One vertical position is reserved for every 5th neuron in the network. The dots indicate the spike times in the initial 10 seconds of the simulation. The neurons are sorted from bottom to top with respect to the magnitude of the external input current. Compare to Fig. 1A of the original publication.

Fig. 2 Histogram of the spiking activity in the network computed as the fraction of neurons emitting an action potential in consecutive intervals of 1 ms. Compare to Fig. 1B of the original publication.

Fig. 3 Sorted spike rates of the excitatory neurons. Compare to Fig. 2A of the original publication.

Neuron model

The point neuron model has leaky integrate-and-fire dynamics with exponentially decaying postsynaptic currents. The dynamics is integrated exactly on the chosen time grid of 0.25 ms. During an absolute refractory period the neuron cannot generate spikes and the membrane potential is clamped to the resting potential. The original work uses an approximative integration scheme.

Each neuron is subject to a constant background current which keeps the membrane potential near threshold. The amplitudes of the currents of the individual neurons are drawn from a uniform distribution with mean I0_dc = V_th / R = 15.0 pA and a range I0_delta = 0.05 V_th / R = 0.75 mv. Thus, for all neurons the current i0 satisfies .

Following a spike the membrane potential is clamped to the reset potential for tau_ref_abs and no spike can be generated for time tau_ref_tot. Thus, if tau_ref_tot > tau_ref_abs there is a period where the membrane potential can evolve but still no spike can be generated.

Synapse model

The dynamics of short term plasticity is governed by the set of differential equations (3)-(5) in (Tsodyks, Uziel, & Markram, 2000). This system of equations is integrated exactly on the time grid of 0.25 ms. Spikes reach their target neuron with a fixed delay of one simulation time step, i.e. 0.25 ms. The original work does not incorporate delays and uses an approximative integration scheme. The parameters are:

U_i(ee)

all distributions are Gaussian. 0.0 ms meaning that the variable describing the probability of release is constant and assumes the value of parameter U.

Initial conditions

The initial membrane potential is drawn individually for neuron from a uniform distribution with values between [0, V_th = 15.0 mV]. The variables representing the state of a synapse (see eq.(3)-(5) in Tsodyks, Uziel, & Markram 2000) are initially set to x = 0.0, y = 0.0, z = 1.0, u = 0.0.

Connectivity

The network is connected randomly. For each neuron, the in-degree C is drawn from a Gaussian distribution with a mean of 50 connections and std = 5. Then 0.8*C (rounded) presynaptic neurons are randomly selected from the pool of excitatory neurons and 0.2*C (rounded) presynaptic neurons are chosen from the pool inhibitory. For each connection the synaptic parameters are chosen as specified in section II except that, in contrast to the original work, the time constants of facilitation and recovery are set to the mean values of the respective distributions.

References

1. Tsodyks M, Uziel A, and Markram H (2000) Synchrony Generation in Recurrent Networks with Frequency-Dependent Synapses. The Journal of Neuroscience 20 RC50:1-5
2. Rotter S, and Diesmann M (1999) Exact Digital Simulation of Time-Invariant Linear Systems with Applications to Neuronal Modeling. Biological Cybernetics 81:381-402