Brette, R. (2006). Exact simulation of integrate-and-fire models with synaptic conductances. Neural Comput 18(8): 2004-2027.
Abstract. Computational neuroscience relies heavily on the simulation of large networks of neuron models. There are essentially two simulation strategies: 1) using an approximation method (e.g. Runge-Kutta) with spike times binned to the time step; 2) calculating spike times exactly in an event-driven fashion. In large networks, the computation time of the best algorithm for either strategy scales linearly with the number of synapses, but each strategy has its own assets and constraints: approximation methods can be applied to any model but are inexact; exact simulation avoids numerical artefacts but is limited to simple models. Previous work has focused on improving the accuracy of approximation methods. In this paper we extend the range of models that can be simulated exactly to a more realistic model, namely an integrate-and-fire model with exponential synaptic conductances.
- Functions for exact simulation: IFSC.h, IFSC.c (to include in your favourite event-driven simulator)
- Scilab implementation of functions for exact simulation and an example script for a random network: ExpCondIF(Brette).zip. The archive includes a generic event-driven simulator written in Scilab for networks without delays (this is for pedagogical purposes and is not intended to be efficient at all). N.B.: Scilab is a free scientific software (resembling Matlab).
Brette, R. (2007). Exact simulation of integrate-and-fire models with exponential currents. Neural Comput 19(10): 2604-2609.
Abstract. Neural networks can be simulated exactly using event-driven strategies, in which the algorithm advances directly from one spike to the next spike. It applies to neuron models for which we have 1) an explicit expression for the evolution of the state variables between spikes and 2) an explicit test on the state variables which predicts whether and when a spike will be emitted. In a previous work, we proposed a method which allows exact simulation of an integrate-and-fire model with exponential conductances, with the constraint of a single synaptic time constant. In this note we propose a method, based on polynomial root finding, which applies to integrate-and-fire models with exponential currents, with possibly many different synaptic time constants. Models can include biexponential synaptic currents and spike-triggered adaptation currents.
- Scilab implementation of functions for exact simulation and an example script for a random network: ScilabExpIF.zip. The archive includes an event-driven simulator written in Scilab for networks with random external events and without delays (this is for pedagogical purposes and is not intended to be efficient at all). N.B.: Scilab is a free scientific software (resembling Matlab).
- C++ implementation of the same example: EventDrivenExpIF.zip (with the algorithm from the paper) and ClockDrivenExpIF.zip (with a standard clock-driven algorithm).
Fontaine, B. and Brette, R. (2011). Neural development of binaural tuning through Hebbian learning predicts frequency-dependent best delays. J Neurosci 31(32):11692–11696
Abstract. Birds use microsecond differences in the arrival times of the sounds at the two ears to infer the location of a sound source in the horizontal plane. These interaural time differences (ITDs) are encoded by binaural neurons which fire more when the ITD matches their "best delay". In the textbook model of sound localization, the best delays of binaural neurons reflect the differences in axonal delays of their monaural inputs, but recent observations have cast doubts on this classical view because best delays were found to depend on preferred frequency. Here we show that these observations are in fact consistent with the notion that best delays are created by differences in axonal delays, provided ITD tuning is created during development through spike-timing-dependent plasticity: basilar membrane filtering results in correlations between inputs to binaural neurons, which impact the selection of synapses during development, leading to the observed distribution of best delays.
Movie 1. Evolution of the synaptic weights of 3 neurons with CF = 2, 4 and 6, when presented with a binaurally delayed white noise with ITD = 167 µs.
Movie 2. Evolution of the best delays of 160 neurons with CF distributed between 2 kHz and 8 kHz, when presented with uncorrelated binaural noise.
Movie 3. Evolution of the best delays of 160 neurons with CF distributed between 2 kHz and 8 kHz, when presented with natural stereo recordings.
Kremer Y, Léger JF, Goodman DF, Brette R, Bourdieu L (2011).Late Emergence of the Whisker Direction Selectivity Map in the Rat Barrel Cortex. J Neurosci 31(29):10689-700.
Abstract. In the neocortex, neuronal selectivities for multiple sensorimotor modalities are often distributed in topical maps thought to emerge during a restricted period in early postnatal development. Rodent barrel cortex contains a somatotopic map for whisker identity, but the existence of maps representing other tactile features has not been clearly demonstrated. We addressed the issue of the existence in the rat cortex of an intra-barrel map for whisker movement direction using in vivo two photon imaging. We discovered that the emergence of a direction map occurs long after all known critical periods in the somatosensory system. This map is remarkably specific, taking a pinwheel form centered near the barrel center and aligned to the barrel cortex somatotopy. We suggest that this map may arise from intra-cortical mechanisms and demonstrate by simulation that the combination of spike-timing-dependent plasticity at synapses between layer 4 and layer 2/3 and realistic pad stimulation is sufficient to produce such a map. Its late emergence long after other classical maps suggests that experience-dependent map formation and refinement continue throughout adult life.
Supplementary movie (20 MB). Spikes of L2/3 neurons and evolution of their direction selectivity during the stimulation with random moving bars. The movie shows the spikes of all neurons in L2/3 during the stimulation with random moving bars (left) and the evolution of their direction selectivity (right), as estimated from the average selectivity of their presynaptic neurons in L4, weighted by the synaptic weights. Initial conditions and parameters are identical to those used in the Figure 6. In these conditions, the simulation lasted 1500 seconds. Only 3% of all frames are included in the movie. Note that when a moving bar activates a sequence of whiskers, the first barrel column reached by the bar is almost homogeneously activated (at least at the beginning of the simulation) whereas subsequent barrel columns already exhibit inhomogeneous activity. This inhomogeneous activation of the second, third, etc barrel columns is not a consequence of the existence of a radial direction selectivity map, as this map does not have enough time to develop for instance at the first presentation of the moving bar. It is just due to horizontal connections within layer 2/3, which propagate the activation of the first column to the next adjacent columns, inhibiting predominantly the closest regions to the first column. This inhomogeneous activation of barrel columns becomes reinforced as the direction selectivity map develops, but this contribution only comes in second.
Code. In the examples of the Brian simulator, there is a script that stimulates a plastic model of the barrel cortex with randomly moving bars, and shows the selectivity map after learning.