## The Spiking Neuron

dendrites
input device
soma
central processing unit (non-linear processing step). If the total input exceeds a certain threshold, an output signal is generated
axon
output device, delivering signal to other neurons
synapse
junction between two neurons
post/presynaptic cells
If a neuron is sending a signal across a synapse, the sending neuron is the presynaptic cell, and the receiving neuron is the postsynaptic cell
action potentials/spikes
short electrical pulses, typically of amplitude about 100mV and a duration of 1-2ms
spike train
a chain of action potentials (sequence of stereotyped events) that occur at intervals. Since all spikes of a given neuron look the same, the form of the spike does not matter: the number and timing of the spikes encode the information.
absolute refractory period
minimal distance between two spikes. Spike are well separated, and it is impossible to excite a second spike within this refractory period.
relative refractory period
follows the absolute refractory period – a period where it is difficult to excite an action potential

We define for presynaptic neuron $$j$$, $$\epsilon_{ij}(t) = u_{i}(t) - u_{rest}$$. For a few input spikes, the membrane potential responds roughly linearly to the input spikes:

\begin{equation} u_i{t} = \sum_{j}\sum_{f} \epsilon_{ij}(t - t_j^{(f)}) + u_{rest} \end{equation}

If $$u_i(t)$$ reaches threshold $$\vartheta$$ from below, neuron $$i$$ fires a spike.

From the above, we can define the Spike Response Model describing the momentary value of the membrane potential of neuron $$i$$:

\begin{equation} u_i{t} = \eta (t - \hat{t_i}) + \sum_{j}\sum_{f} \epsilon_{ij}(t - t_j^{(f)}) + u_{rest} \end{equation}

where $$\hat{t_i}$$ is the last firing time of neuron $$i$$.

We refer to moment when a given neuron emits an action potential as the firing time of that neuron. We denote the firing times of neuron $$i$$ by $$t_i^{(f)}$$ where $$f = 1,2,\dots$$ is the label of the spike. Then we formally denote the spike train of a neuron $$i$$ as the sequence of firing times:

\begin{equation} S_i(t) = \sum_{f} \delta\left( t - t_i^{(f)} \right) \end{equation}

where $$\delta(x)$$ is the Dirac $$\delta$$ function with $$\delta(x) = 0$$ for $$x \ne 0$$ and $$\int_{-\infty}^{\infty} \delta(x)dx = 1$$. Spikes are thus reduced to points in time.

SRM only takes into account the most recent spike, and cannot capture adaptation.

### Neuronal Coding

How do spike trains encode information? At present, a definite answer to this question is not known.

• Temporal Coding

Traditionally, it had been thought that information was contained in the mean firing rate of a neuron:

\begin{equation} v = \frac{n_{sp}(T)}{T} \end{equation}

measured over some time window $$T$$, counting the number of the spikes $$n$$. The primary objection to this is that if we need to compute a temporal average to transfer information, then our reaction times would be a lot slower.

From the point of view of rate coding, spikes are a convenient wa of transmitting the analog output variable $$v$$ over long spikes. The optimal scheme is to transmit the value of rate $$v$$ by a regular spike train at intervals $$\frac{1}{v}$$, allowing the rate to be reliably measured after 2 spikes. Therefore, irregularities in real spike trains must be considered as noise.

• Rate as spike density (average over several runs)

this definition works for both stationary and time-dependent stimuli. The same stimulation sequence is repeated several times, and the neuronal response is reported in a peri-stimulus-time histogram (PSTH). We can obtain the spike density of the PSTH by:

\begin{equation} \rho(t) = \frac{1}{\Delta t} \frac{n_K(t; t + \Delta t)}{K} \end{equation}

where $$K$$ is the number of repetitions of the experiment. We can smooth the results to get a continuous rate.

The problem with this scheme is that it cannot be the decoding scheme of the brain. This measure makes sense if there is always a population of neurons with the same stimulus. This leads to population coding.

• Rate as population activity (average over several neurons)

This is a simple extension of the spike density measure, but adding activity across a population of neurons. Population activity varies rapidly and can reflect changes in the stimulus nearly instantaneously, an advantage over temporal coding. However, it requires a homogeneous population of neurons, which is hardly realistic.

### Spike Codes

These are coding strategies based on spike timing.

• Time-to-first-spike

A neuron which fires shortly after the reference signal (an abrupt input, for example) may signal a strong stimulation, and vice-versa. This estimate has been successfully used in an interpretation of neuronal activity in primate motor cortex.

The argument is that the brain does not have time to evaluate more than one spike per neuron per processing step, and hence the first spike should contain most of the relevant information.

• Phase

Oscillations are common in the olfactory system, and other areas of the brain. Neuronal spike trains could then encode information in the phase of a pulse, with respect to the background oscillation.

• Correlations and Synchrony

Synchrony between any pairs of neurons could signify special events and convey information not contained in the firing rate of the neurons.

### Spikes or Rates?

A code based on time-to-first-spike is consistent with a rate code: if the mean firing rate of a neuron is high, then the time to first spike is expected to occur early. Stimulus reconstruction with a linear kernel can be seen as a special instance of a rate code. It is difficult to draw a clear borderline between pulse and rate codes. The key consideration in using any code is the ability for the system to react quickly to changes in the input. If pulse coding is relevant, information processing in the brain must be based on spiking neuron models. For stationary input, spiking neuron models can be reduced to rate models, but in other cases, this reduction is not possible.

## What are Spiking Neural Networks?

1. Generally naturally recurrent
2. Inspired by biological information processing: mimics how the brain functions

Models of cortical hierarchies have inspired the architectural principles of deep neural networks, but at the implementation level there is only marginally similar to brain-like computation and analog neural networks.

## What’s different between DNNs and SNNs?

1. Artificial neural networks are mostly non-linear but continuous

function approximators that operate on a common clock cycle, while SNNs compute with asynchronous spikes that signal the occurrence of some characteristic event.

## In what areas are SNN better?

They exhibit favourable properties similar to brain circuits:

1. low power consumption
2. analog computation
3. fast inference
4. event-driven processing
5. online learning
6. parallelism

Deep SNNs are also most capable of making use of event-based sensors, that are slowly becoming mature enough for production use. They can utilize efficient temporal codes in their computations. Typically sensory information from the outside world is sparse: this leads to power efficiency.

SNNs also enable pseudo-simultaneous information processing. Even for multi-layer neural networks, spikes begin to propagate immediately to higher layers as soon as the lower layer provides sufficient activity. Initial output spikes are typically composed from incomplete information.

## How does a SNN work? (Michael Pfeiffer & Thomas Pfeil, 2018)

Neurons exchange information via spikes, and the information received depends on:

Firing frequencies
relative timing of pre and post-synaptic spikes, and the neuronal firing patterns
Identity of synapses used
which neurons are connected, whether the synapse is inhibitory or excitatory, synaptic strength etc.

## Difficulties on training SNNs

Conventional deep learning relies on gradient methods, such as stochastic gradient descent. These require differentiable activation functions.

Integrating the timing of spikes requires additional effort.

1. Binarization of ANNs: Conventional DNNs are trained with binary activations, and maintain their synchronous mode of information processing
2. Conversion from ANNs: Conventional DNNs are trained with backpropagation, and then all analog neurons are converted into spiking neurons
3. Training of constrained networks: Before conversion, conventional DNN training methods are used together with constraints that model the properties of SNNs
4. Supervised learning with spikes: Direct training of SNNs using variations of error backpropagation
5. Local learning rules at synapses, such as STDP

### Binarization of ANNs

Binary networks offer efficient inference, at the cost of slight performance degradation. The performance benefits come from efficient multiply-add operations (using XNOR, bit-counting etc.), and low precision computation.

Non-differentiability of the output of the SNN makes it difficult to train SNNs with conventional backpropagation methods. Some learning rules approximate the derivative by smoothing out the membrane potential as a function of the weights.

### Conversion from ANNs

The initial approach was to convert activations of analog neurons into firing rates of spiking ones. The Neural Engineering Framework also allows conversion of RBMs into SNNs. Conversion from ANNs provide the full toolkit of deep learning for training.

However, some ANNs are difficult to convert to SNNs. For example, negative activations are difficult to convert to firing rates, since firing rates are always positive. ReLUs are easier to convert, having activations being only zero or positive.

### Training under SNN constraints

Constrain-then-train includes constraints from the properties of spiking neurons during the training process. Conventional ANN learning methods such as backpropagation are then used to learn the optimal weights under these constraints.

### Supervised Learning with Spikes

The key for many spike-based learning rules for multi-layer SNNs is to find a real-valued and almost-everywhere differentiable proxy, on which backpropagation can be performed. SpikeProp derives a backpropagation rule for spike times in the output layer. Spike-based backpropagation methods exist, such as performing gradient descent on real-valued membrane potentials.

### Local Learning Rules

From a practical perspective, local learning is desirable as it allows for hardware-efficient learning. However, supervised error signals are typically only available in the output layer, and it is unclear how to propagate the errors to lower layers. Hence, recurrent feedback connections are introduced to modulate training in these lower layers.

Local learning rules such as Hebbian learning and Spike-timing-dependent-plasticity (STDP) are have been used to train competitive networks that are able to do clustering.

## Probabilistic SNNs

A probabilistic model defines the outputs of all spiking neurons as jointly distributed binary random processes. The joint distribution is differentiable in the synaptic weights, and principled learning criteria from statistics and information theory such as likelihood and mutual information apply. The maximization of such criteria do not require the implementation of the backpropagation mechanism, and often recover as special cases known biologically plausible algorithms.

## Graphical Representation

A SNN consists of a network of $$N$$ spiking neurons. At any time $$t = 0,1,2, \dots$$ each neouron $$i$$ outputs a binary signal $$s_{i,t} = \{0,1\}$$, with value $$s_{i,t} = 1$$ corresponding to a spike emitted at time $$t$$. We collect in vector $$s_{t} = \left( s_{i,t}: i \in V \right)$$ the binary signals emitted by all neurons at time $$t$$, where $$V$$ is the set of all neurons. Each neuron $$i \in V$$ receives the signals emitted by a subset $$P_i$$ of neurons through directed links, known as synapses. Neurons in a set $$P_i$$ are referred to as pre-synaptic for post-synaptic neuron $$i$$.

The internal, analog state of each spiking neuron $$i \in V$$ at time $$t$$ is defined by its membrane potential $$u_{i,t}$$.

## Long short-term memory and learning-to-learn in networks of spiking neurons (Bellec et al., 2018)

Key contribution: Inclusion of adapting neurons into recurrent SNN models (RSNNs) increases computing and learning capability. By using a learning algorithm that combines BPTT with a rewiring algorithm that optimizes the network architecture, performance comes close to LSTM ANNs.

Model composition: LSNNs consist of a populaction $$R$$ of integrate-and-fire (LIF) neurons (excitatory and inhibitory), and a second population $$A$$ of LIF excitatory neurons whose excitability is temporarily reduced through preceding firing activity. $$R$$ and $$A$$ receive spike trains from a population $$X$$ of external input neurons. Results of computations are read out by a population $$Y$$ of external linear readout neurons.

BPTT is done by replacing the non-continuous membrane potential with a pseudo derivative that smoothly increases from 0 to 1.

### Learning to Learn LSNNs

LSTM networks are especially suited for L2L since they can accommodate two levelsof learning and representation of learned insight: Synaptic connections and weights can encode,on a higher level, a learning algorithm and prior knowledge on a large time-scale. The short-termmemory of an LSTM network can accumulate, on a lower level of learning, knowledge during thecurrent learning task

## Gradient Descent for Spiking Neural Networks (Huh & Sejnowski, 2017)

key idea: Replacing the non-differentiable model for membrane potential:

\begin{equation} \tau \dot{s} = -s + \sum_{k} \delta (t - t_k) \end{equation}

with

\begin{equation} \tau \dot{s} = -s + g \dot{v} \end{equation}

for some gate function $$g$$, and $$\dot{v}$$ is the time derivative of the pre-synaptic membrane voltage.

Exact gradient calculations can be done with BPTT, or real-time recurrent learning. The resultant gradients are similar to reward-modulated spike-time dependent plasticity.

## STDP

STDP is a biologically inspired long-term plasticity model, in which each synapse is given a weight $$0 \le w \le w_{maxx}$$ , characterizing its strength, and its change depends on the exact moments $$t_{pre}$$ of presynaptic spikes and $$t_{post}$$ of postsynaptic spikes:

\begin{equation} \Delta w=\left\{\begin{array}{l}{-\alpha \lambda \cdot \exp \left(-\frac{t_{\mathrm{pre}}-t_{\mathrm{post}}}{\tau_{-}}\right), \text {if } t_{\mathrm{pre}}-t_{\mathrm{post}}>0} \ {\lambda \cdot \exp \left(-\frac{t_{\mathrm{post}}-t_{\mathrm{pre}}}{\tau_{+}}\right), \text {if } t_{\mathrm{pre}}-t_{\mathrm{post}}<0}\end{array}\right. \end{equation}

This additive STDP rule requires also an additional constraint that explicitly prevents the weight from falling below 0 or exceeding the maximum value of 1.

(Alexander Sboev et al., 2018)

## Loihi

• Describes SNNs as a weighted, directed graph $$G(V, E)$$ where the vertices $$V$$ represent compartments, and the weighted edges $$E$$ represent synapses.
• Both compartments and synapses maintain internal state and communicate only via discrete spike impulses.
• Uses a variant of the CUBA model for the neuron model, which is defined as a set of first-order differential equation using traces, evaluated at discrete algorithmic time steps.

Learning must follow the sum-of-products form:

\begin{equation} Z(t) = Z(t-1) + \sum_m S_m \prod_n F_n \end{equation}

where $$Z$$ is the synaptic state variable defined for the source destination neuron pair being updated, and $$F-N$$ may be a synaptic state variable, a pre-synaptic trace or a post-synaptic trace defined for the neuron pair.

## Generating Spike Trains

### Poisson Model (Heeger, 2000)

Independent spike hypothesis: the generation of each spike is independent of all other spikes. If the underlying instantaneous firing rate $$r$$ is constant over time, it is a homogeneous Poisson process.

We can write:

\begin{equation} P(\textrm{1 spike during } \delta t) \approx r \delta t \end{equation}

We divide time into short, discrete intervals $$\delta t$$. Then, we generate a sequence of random numbers $$x[i]$$ uniformly between 0 and 1. For each interval, if $$x[i] \le r \delta t$$, generate a spike.

# Bibliography

Pfeiffer, M., & Pfeil, T., Deep learning with spiking neurons: opportunities and challenges, Frontiers in Neuroscience, 12(nil), (2018). http://dx.doi.org/10.3389/fnins.2018.00774

Bellec, G., Salaj, D., Subramoney, A., Legenstein, R., & Maass, W., Long short-term memory and learning-to-learn in networks of spiking neurons, CoRR, (), (2018).

Huh, D., & Sejnowski, T. J., Gradient descent for spiking neural networks, CoRR, (), (2017).

Neftci, E. O., Mostafa, H., & Zenke, F., Surrogate gradient learning in spiking neural networks, CoRR, (), (2019).

Whittington, J. C., & Bogacz, R., Theories of error back-propagation in the brain, Trends in Cognitive Sciences, 23(3), 235–250 (2019). http://dx.doi.org/10.1016/j.tics.2018.12.005

Sboev, A., Vlasov, D., Rybka, R., & Serenko, A., Spiking neural network reinforcement learning method based on temporal coding and stdp, Procedia Computer Science, 145(nil), 458–463 (2018). http://dx.doi.org/10.1016/j.procs.2018.11.107

Heeger, D., Poisson model of spike generation, Handout, University of Standford, 5(), 1–13 (2000).