# Smoothed Spiking Neural Networks

Smoothed Spiking Neural Networks ensure well-behaved gradients which are directly suitable for optimization. They come in 4 categories:

1. Soft non-linearity models
2. Probabilistic models, where gradients are well defined in expectation
3. Models relying on rate or
4. Single-spike temporal codes

## Soft non-linearity models

This approach can be applied on all spiking neuron models which include a smooth spike generating process. These include:

• Hodgkin-Huxley
• Morris-Lecar
• FitzHugh-Nagumo

The resultant network can be optimized using standard methods of BPTT. These smoothed models compromise on a key feature of SNNs: binary spike propagation.

## Probabilistic Models

Stochasticity in the models effectively smooths out the discontinuous binary nonlinearity, which provides well-defined gradients on expectation values. These binary probabilistic models are commonly studied in Restricted Boltzmann machines.

With probabilistic models, the log-likelihood of a spike train is a smooth quantity, which can be optimized using gradient-descent.

## Gradients in rate-coding networks

In rate-coding networks, it is assumed that the spike rate carries the underlying information. Rate-based approaches offer good performance in practice, but may be inefficient. Precise estimation of firing rates require averaging over a number of spikes. These require high firing rates, or longer averaging times.

Probabilistic network implementations also use rate-coding at the output level.

## Single-spike-time-coding networks

In the temporal coding setting, individual spikes carry significantly more information. This was first pioneered by SpikeProp. Firing times for hidden units were linearized, allowing to analytically compute approximate hidden layer gradients.