Building functional networks of spiking model neurons

Abstract

Most of the networks used by computer scientists and many of those studied by modelers in neuroscience represent unit activities as continuous variables. Neurons, however, communicate primarily through discontinuous spiking. We review methods for transferring our ability to construct interesting networks that perform relevant tasks from the artificial continuous domain to more realistic spiking network models. These methods raise a number of issues that warrant further theoretical and experimental study.

Figure 1

Structure of autonomous and driven networks. (a) The autonomous network. In this diagram, black lines and dots denote fixed connections, and red lines and dots are connections that are adjusted to make the network function properly. A defined input f_in is provided to the network through connections characterized by weights u. Neurons in the network are connected by two types of synapses, parameterized by J_fast (in black) and J (in red). The problem is to choose the strengths of the synapses defined by J, and the weights w, so that the output of the network, ws, approximates a given target output f_out. (b) The driven network. In this case, the network is driven by input f_D, through weights u_D, that forces it to produce the desired output. Only the fixed synapses denoted by J_fast are included. Output weights are adjusted as in the autonomous network.

Figure 2

Driven networks approximating a continuous target output. (a) Spike train from a single model neuron (top), the normalized synaptic current s(t) that it generates (middle) and the output ws computed from a weighted sum of the normalized synaptic currents from this neuron and 99 others (bottom). (b–d) Results from driven networks with optimally tuned readout weights. In each, the upper plot shows the actual output ws in red and the target output f_out in black, and the lower plot shows representative membrane potential traces for 8 of the 1,000 integrate-and-fire model neurons in each network. Neurons in the driven network are connected by fast synapses with random weights for b and c and with weights adjusted according to the spike-coding scheme for d. The three panels show the outputs in response to a driving input f_D = f_out (b), a driving input f_D = f_out + τdf_out/dt in a rate-coding network (c) and a driving input f_D = f_out + τdf_out/dt in a spike-coding network (d).

Figure 3

Two autonomous networks of spiking neurons constructed to integrate the input f_in (top, black traces). (a) A rate-coding network. (b) A spike-coding network. For each network, the results from two trials are shown. The upper red and blue traces marked ws show the output of the networks on these two trials (they overlap almost perfectly in b and are therefore difficult to distinguish), and the bottom blue and red traces show the membrane potentials of three neurons in the networks on the two trials. Note the trial-to-trial variability in the spiking patterns. Each network consists of 1,000 model neurons.

Figure 4

Autonomous networks solving a temporal XOR task. (a) A rate-coding network with linear neuronal input integration. (b) A spike-coding network with nonlinear neuronal input integration. In both cases, the network output (red traces) is a delayed positive deflection if two successive input pulses have different signs and is a negative deflection if the signs are the same. Blue traces show the membrane potentials of four neurons in the networks.