Building and testing a spiking network model

A network model is an experiment done in a computer. Instead of an animal or a slice, the preparation is a set of equations: one for each neuron and one for each connection. The modeller chooses the cells, the wiring and the strengths of the synapses, runs the simulation, records every spike, and analyses the result with the same measures used for real recordings. Models are used to ask how the activity of a circuit arises from its cells and connections, and to make predictions that can be tested with recordings (in silico work).

The neurons in this session follow the simple model of Izhikevich, which reproduces the firing of several types of cortical neuron with two equations and four parameters, and is efficient enough to simulate a thousand neurons at a time step of 1 ms in a browser (Izhikevich, 2003). About four in five cells are excitatory, like pyramidal neurons; the rest are inhibitory interneurons that release GABA. Brunel showed that sparsely connected networks of excitatory and inhibitory neurons can be asynchronous and irregular, synchronous and regular, or oscillatory, depending on the external input and on the balance between excitation and inhibition (Brunel, 2000).

The work is done at the modeller's desk, but the logic is that of the laboratory. Each random network built by the same rules is one sample, so a result is repeated on several networks (random seeds), as an experiment is repeated on several animals. Conclusions are checked for robustness, and the model must be compared with recordings before it is trusted. The model has a coarse time step of 1 ms, and its state labels use simple thresholds chosen for this site.

Learning objectives

• Write down the Izhikevich model neuron and test a parameter set against a recording by its firing rate, its frequency–current curve and its adaptation
• Build a random network of excitatory and inhibitory populations, draw its connectivity and calculate how many inputs each cell receives
• Set the synaptic weights so that the summed excitation and inhibition match a reference network, and predict what changing the balance will do
• Read a raster plot, a population rate and a power spectrum, measure the frequency of a rhythm and tell asynchronous from synchronous, oscillatory and seizure-like activity
• Design a parameter sweep, repeat it on independent random networks, and summarise it with a slope and confidence interval across seeds
• Compare model output with published results, say what recordings would test its predictions, and state clearly what the model can and cannot show

Ethics and safety

Modelling needs no animals of its own, but the data used to build and test a model usually come from animal or human experiments, which require approval by an ethics committee (in India, an Institutional Ethics Committee for human research and an Institutional Animal Ethics Committee for animal research) and, for animals, should follow the 3Rs (replacement, reduction and refinement). Models can help to reduce animal use by focusing experiments on the most informative conditions; they do not replace the data against which they must be validated. Good modelling practice is to write down the model, every parameter value and the hypothesis before the simulations, to repeat each condition on several random networks, and to share code, parameters and seeds so that others can reproduce every figure.

Key terms

• Model neuron: A set of equations for the membrane potential of one cell. Here the Izhikevich model: v′ = 0.04v² + 5v + 140 − u + I and u′ = a(bv − u), with v reset to c and u increased by d after each spike at 30 mV.
• Regular spiking and fast spiking: Two firing types. Regular-spiking cells (most pyramidal neurons) slow down during a long current step (adaptation); fast-spiking interneurons fire at high rates with little slowing.
• Frequency–current (f–I) curve: The firing rate of a cell plotted against the size of the injected current step; a standard test of a model neuron against recordings.
• Connection probability: The chance that any one cell sends a synapse to any other; with 1000 cells and a probability of 0.1, each cell receives about 100 inputs.
• Synaptic weight: How much one spike of the presynaptic cell changes the input to the postsynaptic cell. Here the summed strength J is the expected total weight a cell receives from a whole population.
• Excitation–inhibition balance: The relation between the excitatory and the inhibitory input to the cells. See excitation and inhibition.
• Raster plot: A plot with one row per neuron and a dot at each spike time; vertical stripes show many cells firing together.
• Population rate: The fraction of cells firing in each short time window, expressed in spikes per second (Hz) per cell.
• LFP-like signal: A stand-in for the local field potential: here the mean, over the excitatory cells, of the size of their excitatory plus their inhibitory synaptic input.
• Power spectrum: How the power of a signal is spread over frequencies. A peak marks a rhythm; its position is the rhythm's frequency. See brain rhythm.
• Synchrony index χ²: The variance of the population rate divided by the mean variance of single cells' spike counts: about 1/N for independent cells and 1 for perfect synchrony.
• CV of inter-spike intervals: The standard deviation of the intervals between a cell's spikes divided by their mean: near 0 for clock-like firing and about 1 for random (Poisson) firing.
• Random seed: The starting number of the random-number generator. A different seed builds a different network by the same rules; seeds play the role of replicate animals. See replicates.
• Parameter sweep: A set of simulations in which one parameter is changed step by step while everything else stays fixed.