Exact Subthreshold Integration with Continuous Spike Times in Discrete-Time Neural Network Simulations

Abstract

Very large networks of spiking neurons can be simulated efficiently in parallel under the constraint that spike times are bound to an equidistant time grid. Within this scheme, the subthreshold dynamics of a wide class of integrate-and-fire-type neuron models can be integrated exactly from one grid point to the next. However, the loss in accuracy caused by restricting spike times to the grid can have undesirable consequences, which has led to interest in interpolating spike times between the grid points to retrieve an adequate representation of network dynamics. We demonstrate that the exact integration scheme can be combined naturally with off-grid spike events found by interpolation. We show that by exploiting the existence of a minimal synaptic propagation delay, the need for a central event queue is removed, so that the precision of event-driven simulation on the level of single neurons is combined with the efficiency of time-driven global scheduling. Further, for neuron models with linear subthreshold dynamics, even local event queuing can be avoided, resulting in much greater efficiency on the single-neuron level. These ideas are exemplified by two implementations of a widely used neuron model. We present a measure for the efficiency of network simulations in terms of their integration error and show that for a wide range of input spike rates, the novel techniques we present are both more accurate and faster than standard techniques.

Simulation of networks of spiking neurons: A review of tools and strategies

“…However, in principle, it is possible to simulate a network exactly in a clock-driven fashion when the minimum transmission delay is larger than the time step. It implies that the precise timing of synaptic events is stored in the circular array (as described in Morrison et al 2007a). Then within each time bin, synaptic events for each neuron are sorted and processed in the right order, and when the neuron spikes, the exact spike timing is calculated.…”

“…In all cases, this quest leads to the implementation of a specific strategy for numerical simulations which is found to be optimal given the set of constraints set by the particular simulation tool. However, as shown recently (Hansel et al 1998;Lee and Farhat 2001;Morrison et al 2007a), quantitative results and their qualitative interpretation strongly depend on the simulation strategy utilized, and may vary across available simulation tools or for different settings within the same simulator. The specificity of neuronal simulations is that spikes induce either a discontinuity in the dynamics (IF models) or have very fast dynamics (HH type models).…”

“…We have recently shown that both parts of this perception are generally false; it depends on the dynamics of the neuron model whether an event-driven algorithm can find an exact solution, just as it does for time-driven algorithms (Morrison et al 2007a). NEST is designed for large scale simulations where performance is a critical issue.…”

“…We have therefore argued that when comparing different integration strategies, one should evaluate the efficiency, i.e. the simulation time required to achieve a given integration error, rather than the plain simulation time (Morrison et al 2007a). This philosophy is reflected in the simulation kernel of NEST.…”

“…In the case of neuron models where the synaptic time course is not invertible, the computational time step determines the accuracy of the calculation of the threshold crossing. For a discussion of this case and the relevant quantitative benchmarks, see Morrison et al (2007a).…”