Brian: a simulator for spiking neural networks in python

Abstract

"Brian" is a new simulator for spiking neural networks, written in Python (http://brian. di.ens.fr). It is an intuitive and highly flexible tool for rapidly developing new models, especially networks of single-compartment neurons. In addition to using standard types of neuron models, users can define models by writing arbitrary differential equations in ordinary mathematical notation. Python scientific libraries can also be used for defining models and analysing data. Vectorisation techniques allow efficient simulations despite the overheads of an interpreted language. Brian will be especially valuable for working on non-standard neuron models not easily covered by existing software, and as an alternative to using Matlab or C for simulations. With its easy and intuitive syntax, Brian is also very well suited for teaching computational neuroscience.

Keywords: Python; computational neuroscience; integrate and fire; neural networks; simulation; software; spiking neurons; teaching.

Figure 1

The CUBA network in Brian, with code on the left, neuron model equations at the top right and output raster plot at the bottom right. This script defines a randomly connected network of 4000 leaky integrate-and-fire neurons with exponential synaptic currents, partitioned into a group of 3200 excitatory neurons and 800 inhibitory neurons. The subgroup() method keeps track of which neurons have been allocated to subgroups and allocates the next available neurons. The process starts from neuron 0, so Pe has neurons 0 through 3199 and Pi has neurons 3200 through 3999. The script outputs a raster plot showing the spiking activity of the network for a few hundred ms. This is Brian's implementation of the current-based (CUBA) network model used as one of the benchmarks in Brette et al. (2007), based on the network studied in Vogels and Abbott (2005). The simulation takes 3–4 s on a typical PC (1.8 GHz Pentium), for 1 s of biological time (with dt = 0.1 ms). The variables g_e and g_i are not conductances, we follow the variable names used in Brette et al. (2007). The code :volt in the equations means that the unit of the variable being defined (V, g_e and g_i) has units of volts.

Figure 2

An example showing many of the features of Brian in action. The neuron model in this code follows a stochastic differential equation $\text{d}V/\text{d}T=-\text{(}V-E_l\text{)}/\tau \text{+}\sigma \xi \text{(}t\text{)}/\sqrt{\tau }$, dV_t/dt = −(V_t − V_t0)/τ_t. Here all the undefined symbols are constants except for ≈(t) which corresponds to the term xi in the code, and represents a white noise term $\left(\langle \xi \text{(}t\text{)}\xi \text{(}{t}^{\prime }\text{)}\rangle =\delta \text{(}t-t^{\prime }\text{)}\right)$. The rest of the neuron model is defined by a custom reset function adaptive_threshold_reset which increases the value of V_t by a constant each time a neuron spikes (but never takes it above a fixed ceiling), and a custom threshold function lambda V,Vt:V>=Vt which defines the condition for a spike. The arguments to the custom reset function are a NeuronGroup object P (a population of neurons), and an array spikes containing the indices of the neurons in P that have spiked. Together these two custom functions define an adaptive threshold model. The option to specify custom functions makes Brian's reset and threshold mechanism very flexible. The code also shows synaptic delays, and setting the synaptic weights with a custom function of (i, j), w*cos(2.*pi*(i-j)*1./100)). The output of the code shown is the raster plot in (B), with the value w=.5*mV. (A) shows w=.1*mV and (C) shows w=.65*mV. (D) shows the synaptic weight matrix for the w=.65*mV case. (E) and (F) show the values of V (solid blue) and Vt (dashed green) for the neuron with index 50 for the raster plots immediately above them ((B) and (C)) with w=.5*mV and w=.65*mV respectively.

Figure 3

The code from Figure 1 expanded to show how Brian works internally. In (A), the equations for the model are defined. In (B), a group of 4000 neurons is created with these equations. In (C), the logical structure of the network is defined, partitioning the 4000 neurons into excitatory and inhibitory subgroups with corresponding connections to the whole group. In (D), the weight matrices for the excitatory and inhibitory connections are defined. In (E), the membrane potential is initialised uniformly randomly between reset and threshold values. In (F), the simulation is running, consisting of repeated applications of four operations each time step; (a) shows the update of the state matrix; (b) shows the thresholding operation; (c) shows the propagation of spikes; and (d) shows the reset operation.

Figure 4

Computation time for the CUBA network using Brian, C and Matlab. This version of the CUBA network uses a fixed 80 synapses per neuron, and a varying number of neurons N. The figure on the left shows the absolute time on the test machine. The figure on the right shows the time compared to the C code. Theoretically, we would expect O(N) computation time (see Performance of Vectorised Simulations).

Figure 5

Computation time for the CUBA network if all synapses are removed. This largely demonstrates the performance for the state update step, which in this case is a matrix multiplication.

Figure 6

Computation time for the CUBA network with on average p = 500 synapes per neuron and N = 4000 at different firing rates. The parameter w_e, the excitatory weight, was varied between 1.62 and 4.8 mV which had the effect of varying the firing rate between about 5 Hz and about 25 Hz. This shows how performance scales with the number of spikes. Here the firing rates as well as the times are averaged over the seven fastest trials, as firing rates vary from trial to trial. Note that times due to spiking depend on both the firing rate and the number of synapses per neuron.