## Introduction to Spiking Neural Networks

While the project is equal part reinforcement learning and spiking neural networks, reinforcement learning is a popular field and has been extensively covered by researchers worldwide (Ivanov & D’yakonov, 2019), (Li, 2018). Hence, I have chosen instead to review the literature around spiking neural networks.

### The Generations of Neural Networks

Neural network models can be classified into three generations, according to their computational units: perceptrons, non-linear units, and spiking neurons (“Wolfgang Maass”, 1997).

Perceptrons can be composed to produce a variety of models, including Boltzmann machines and Hopfield networks. Non-linear units are currently the most widely used computational unit, responsible for the explosion of progress in machine learning research, in particular, the success of deep learning. These units traditionally apply differentiable, non-linear activation functions such across a weighted sum of input values.

There are two reasons second-generation computational units have seen so much success. First, the computational power of these units is greater than that of first-generation neural networks. Networks built with second-generation computational units with one hidden layer are universal approximators for any continuous function with a compact domain and range (“Cybenko, 1989). Second, networks built with these units are trainable with well-researched gradient-based methods, such as backpropagation.

The third generation of neural networks use computational units called spiking neurons. Much like our biological neurons, spiking neurons are connected to each other at synapses, receiving incoming signals at the dendrites and sending spikes to other neurons via the axon. Each computational unit stores some state: in particular, it stores its membrane potential at any point in time. Rather than fire at each propagation cycle, these computational units fire only when their individual membrane potentials crosses its firing threshold. A simple spiking neuron model is given in <spike_model>.

From this section onwards, we shall term second-generation neural networks Artificial Neural Networks (ANNs), and third-generation neural networks Spiking Neural Networks (SNNs).

### A Spiking Neuron Model

In spiking neural networks, neurons exchange information via spikes, and the information received depends on:

- Firing frequencies
- The relative timing of pre and post-synaptic spikes, and neuronal firing patterns
- Identity of synapses used
- Which neurons are connected, whether their synapses are inhibitory or excitatory, and synaptic strength

Each neuron has a corresponding model that encapsulates its state: the current membrane potential. As with the mammalian brain, incoming spikes increase the value of membrane potential. The membrane potential eventually decays to resting potential in the absence of spikes. These dynamics are often captured via first-order differential equations. Here we define the Spike Response Model (SRM), a simple but widely-used model describing the momentary value of a neuron \(i\).

We define for presynaptic neuron \(j\), \(\epsilon_{ij}(t) = u_{i}(t) - u_{\text{rest}}\). 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_{\text{rest}} \end{equation}

SRM describes the membrane potential of neuron \(i\) as:

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

- 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.

### Motivating Spiking Neural Networks

Since second-generation neural networks have excellent performance, why bother with spiking neural networks? In this section, we motivate spiking neural networks from various perspectives.

- Information Encoding

```
To directly compare ANNs and SNNs, one can consider the real-valued
outputs of ANNs to be the firing rate of a spiking neuron in steady
state. In fact, such rate coding has been used to explain
computational processes in the brain <a id="ffec32504761d0b966ab6c36eee992a6" href="#pfeiffer2018deep">(Pfeiffer \& Pfeil, 2018)</a>. Spiking
neuron models encode information beyond the average firing rate: these
models also utilize the relative timing between spikes
<a id="b77caeaea9e23a2bc9c87b5f10a91487" href="#guetig14_to_spike_or_when_to_spike">(Robert G\"utig, 2014)</a>, or spike phases (in-phase or
out-of-phase). These time-dependent codes are termed temporal codes,
and play an important role in biology. First, research has shown that
different actions are taken based on single spikes
<a id="b7885fc5161a6ad3286ee312ffcc0c0b" href="#stemmler96_singl_spike_suffic">(Martin Stemmler, 1996)</a>. Second, relying on the average firing rate
would greatly increase the latency of the brain, and our brain often
requires decision-making long before several spikes are accumulated.
It has also been successfully demonstrated that temporal coding
achieves competitive empirical performance on classification tasks for
both generated datasets, as well as image datasets like MNIST and
CIFAR <a id="caaddec51f6948e5fea79b6d41c79676" href="#comsa19_tempor_codin_spikin_neural_networ">(Comsa et al., 2019)</a>.
```

```
A faction of the machine learning and neurobiology community strives
for emulation of the biological brain. There are several
incompatibilities between ANNs and the current state of neurobiology
that are not easily reconciliated.
First, neurons in ANNs communicate via continuous-valued activations.
This is contrary to neurobiological research, which shows that
communication between biological neurons communicate by broadcasting
spike trains: trains of action potentials to downstream neurons. The
spikes are to a first-order approximation of uniform amplitude, unlike
the continuous-valued activations of ANNs.
Second, backpropagation as a learning procedure also presents
incompatibilities with the biological brain <a id="cf477ffda17db950a7fd58c9d833182b" href="#TAVANAEI201947">("Amirhossein Tavanaei et al., 2019)</a>.
Consider the chain rule in backpropagation:
\begin{equation} \label{chainrule}
\delta\_{j}^{\mu}=g^{\prime}\left(a\_{j}^{\mu}\right) \sum\_{k} w\_{k j} \delta\_{k}^{\mu}
\end{equation}
\\(\delta\_{j}^{\mu}\\) and \\(\delta\_{k}^{\mu}\\) denote the partial
derivatives of the cost function for input pattern \\(\mu\\) with respect
to the net input to some arbitrary unit \\(j\\) or \\(k\\). Unit \\(j\\) projects
feed-forward connections to the set of units indexed by \\(k\\).
\\(g(\cdot)\\) is the activation function applied to the net input of unit
\\(j\\), denoted \\(a\_j^{\mu}\\), \\(w\_{kj}\\) are the feedforward weights
projecting from unit \\(j\\) to the set of units indexed by \\(k\\).
The chain rule formulation presents two problems. First, the
gradients \\(g'(\cdot)\\) requires derivatives, but \\(g(\cdot)\\) in spiking
neurons is represented by sum of Dirac delta functions, for which
derivatives do not exist. Second, the expression \\(\sum\_{k} w\_{k j}
\delta\_{k}^{\mu}\\) uses feedforward weights in a feedback fashion. This
mean that backpropagation is only possible in the presence of
symmetric feedback weights, but these do not exist in the brain. In
addition, during backpropagation the error assignment for each neuron
is computed using non-local information.
```

```
In a traditional Von Neumann architecture, the logic core operates on
data fetched sequentially from memory. In contrast, in neuromorphic
chips both computation and memory are distributed across computational
units that are connected via synapses. The neuronal architecture and
parameters hence play a key role in information representation and
define the computations that are performed.
It has also been observed that spike-trains in the mammalian brain are
often sparse in time, suggesting that timing and relative timings of
spikes encode large amounts of information. Neuromorphic chips
implement this same sparse, low-precision communication protocol
between neurons on the chip, and by offering the same asynchronous,
event-based parallelism paradigm that the brain uses, are able to
perform certain workloads with much less power than Von Neumann chips.
These integrated circuits are typically programmed with spiking neural
networks. Examples of such chips include IBM's TrueNorth
<a id="ff5e58b063893165385318f053dc98c9" href="#Merolla668">(Merolla et al., 2014)</a> and Intel's Loihi <a id="8146155533911feae2a67ddf637b1e29" href="#davies2018loihi">(Davies et al., 2018)</a>. Because
spiking neural networks have not yet been successfully trained on many
tasks, neuromorphic chips has seen little practical use. These chips
have only recently been successfully used in robotic navigation
<a id="162706e1b73312c94623c3c6304f9856" href="#SnnSlam">("Tang et al., 2019)</a>, and solving graph problems by manual construction of the
network graph <a id="022dd7a783479dfb761bf88b30399ee3" href="#Severa2016SpikingNA">("William Severa et al., 2016)</a>.
```

### Training Spiking Neural Networks

As explained in

- Non-gradient based methods

```
Spiking neurons communicate via spikes, hence, unlike ANNs, gradients
are non-existent. In addition, backpropagation is not biologically
plausible (see <bioplausible>). This motivates the use of
plasticity-based methods and evolutionary strategies for training
SNNs.
One category of learning rules used in SNNs are local learning rules.
These rules include Hebbian learning (neurons that fire together wire
together), and its extension: the spike-timing-dependent-plasticity
rule (STDP). Inspired by experiments in neuroscience, central to these
learning rules is the theme that neuron spike ordering and their
relative timings encode information. STDP adjusts the strength of
connections between neurons using the relative timing of a neuron's
output and its input potentials (hence, spike-timing dependent).
In machine learning terminology, the weights of the synapses are
adjusted according to fixed rules for each training example. Each
synapse is given a weight \\(0 \le w \le w\_{max}\\) , characterizing its
strength, and its change depends on the exact moments \\(t\_{pre}\\) of
pre-synaptic spikes and \\(t\_{post}\\) of post-synaptic spikes
<a id="5c8bcfe50f375b189e564db4d78fe1a3" href="#sboev18_spikin_neural_networ_reinf_learn">(Alexander Sboev et al., 2018)</a>:
\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}
where \\(\tau\_{+}\\) and \\(\tau\_{-}\\) are time constants. \\(\tau\_{+} = 16.8ms\\)
and \\(\tau\_{-} = 33.7ms\\) are reasonable approximations obtained
experimentally.
There are several libraries like BindsNET
<a id="b266899abe9f6133f1e6d55ad24513db" href="#10.3389/fninf.2018.00089">(Hazan et al., 2018)</a> that simulate SNNs on Von Neumann
computers implementing these rules. Recent attempts have been made to
combine Reinforcement Learning and STDP: both in solving RL problems
<a id="b266899abe9f6133f1e6d55ad24513db" href="#10.3389/fninf.2018.00089">(Hazan et al., 2018)</a>, and using the reinforcement learning
framework to train SNN
<a id="54e69da24e936643028e5c2591a092e6" href="#10.3389/fnbot.2019.00018">(Bing et al., 2019)</a><a>, </a><a id="cb84873c6f16a361dfb01de00ae1425a" href="#10.3389/fnins.2018.00435">(Lee et al., 2018)</a>. However, SNNs
trained using the STDP learning rule have yet to achieve comparable
performance compared to ANNs on relatively simple datasets like MNIST
<a id="cf477ffda17db950a7fd58c9d833182b" href="#TAVANAEI201947">("Amirhossein Tavanaei et al., 2019)</a>.
```

- Gradient-based methods

```
Performance is important for practical applications, and
gradient-based training methods such as backpropagation has shown
competitive performance. It is thus desirable to train spiking neural
networks with these gradient-based methods.
There are several problems with spike-compatible gradient-based
methods. First, most of these methods cannot train neurons in the
hidden layers: they can only train neurons at the final layer, that
receive the desired target output pattern
<a id="78cca07dd56a8c50ebd16d663c3b095d" href="#urbanczik09_gradien_learn_rule_tempot">(Robert Urbanczik \& Walter Senn, 2009)</a><a>, </a><a id="9a5e9b63107e695a46943df33a811909" href="#training_deep_snn_bpp_lee">(Lee et al., 2016)</a>.
Second, the discontinuous, binary nature of spiking output needs to be
addressed. For example, SpikeProp approximates the membrane
threshold function at a local area with a linear function, introducing
gradients and computing the exact formulae for error backpropagation
for synaptic weights and spike times <a id="41ed49a4572e3909921301127000035c" href="#spikeprop">(Bohte et al., 2000)</a>. Others have
modified the threshold function with a gate function
<a id="2b991fd4ef5412f89154985cec53e4e8" href="#NIPS2018_7417">(Huh \& Sejnowski, 2018)</a>, used the alpha transfer function to derive
gradient update rules <a id="caaddec51f6948e5fea79b6d41c79676" href="#comsa19_tempor_codin_spikin_neural_networ">(Comsa et al., 2019)</a>,
and approximate the dirac-delta spikes with a probability density
function <a id="123af3c690cf296b196d898878568449" href="#NIPS2018_7415">(Shrestha \& Orchard, 2018)</a>.
Another approach is converting trained ANN models into SNNs
<a id="f842b3cf76d4cc279ae01a4cf75c48ba" href="#rueckauer16_theor_tools_conver_analog_to">(Rueckauer et al., 2016)</a>. Common ANN layers such
as softmax, batch normalization and max-pooling layers have their
corresponding spiking counterparts.
Equilibrium Propagation was recently proposed to solve the
neurobiological incompatibilities of backpropagation
<a id="38c93c265031c562bcf3ab3a89b9d896" href="#10.3389/fncom.2017.00024">(Scellier \& Bengio, 2017)</a>. Because the gradients are defined only
in terms of local perturbations, the synaptic updates correspond to
the standard form of STDP. The propagated signal encodes the gradients
of a well-defined objective function on energy-based models, where the
goal is to minimize the energy of the model. To resolve the issue of
communication using binary-valued signals, step-size annealing was
used to train spiking neural networks with Equilibrium Propagation
<a id="d1886fb7b4cde1636a730b6072eb45c5" href="#pmlr-v89-o-connor19a">(O'Connor et al., 2019)</a>.
```

- Future Research Areas

```
A nascent area is local learning on neuromorphic chips. Thus far
spiking neural networks are simulated and trained before deployment on
a neuromorphic chip. In Intel's Loihi chip, each core contains a
learning engine that can update synaptic weights using the 4-bit
microcode-programmed learning rules that are associated with that
synapse. This opens up areas for online learning.
Neural network models can be classified into three generations,
according to their computational units: perceptrons, non-linear
units, and spiking neurons <a id="622a28118bfc276eace4e2997e32387a" href="#MAASS19971659">("Wolfgang Maass", 1997)</a>.
Perceptrons can be composed to produce a variety of models, including
Boltzmann machines and Hopfield networks. Non-linear units are
currently the most widely used computational unit, responsible for the
explosion of progress in machine learning research, in particular, the
success of deep learning. These units traditionally apply
differentiable, non-linear activation functions such across a weighted
sum of input values.
There are two reasons second-generation computational units have seen
so much success. First, the computational power of these units is
greater than that of first-generation neural networks. Networks built
with second-generation computational units with one hidden layer are
universal approximators for any continuous function with a compact
domain and range <a id="6e22be1723df8ebb3d0e65e120ff8b92" href="#Cybenko1989">("Cybenko, 1989)</a>. Second, networks built with these
units are trainable with well-researched gradient-based methods, such
as backpropagation.
The third generation of neural networks use computational units called
spiking neurons. Much like our biological neurons, spiking neurons are
connected to each other at synapses, receiving incoming signals at the
dendrites and sending spikes to other neurons via the axon. Each
computational unit stores some state: in particular, it stores its
membrane potential at any point in time. Rather than fire at each
propagation cycle, these computational units fire only when their
individual membrane potentials crosses its firing threshold. A simple
spiking neuron model is given in <spike_model>.
From this section onwards, we shall term second-generation neural
networks Artificial Neural Networks (ANNs), and third-generation
neural networks Spiking Neural Networks (SNNs).
```

### A Spiking Neuron Model

In spiking neural networks, neurons exchange information via spikes, and the information received depends on:

- Firing frequencies
- The relative timing of pre and post-synaptic spikes, and neuronal firing patterns
- Identity of synapses used
- Which neurons are connected, whether their synapses are inhibitory or excitatory, and synaptic strength

Each neuron has a corresponding model that encapsulates its state: the current membrane potential. As with the mammalian brain, incoming spikes increase the value of membrane potential. The membrane potential eventually decays to resting potential in the absence of spikes. These dynamics are often captured via first-order differential equations. Here we define the Spike Response Model (SRM), a simple but widely-used model describing the momentary value of a neuron \(i\).

We define for presynaptic neuron \(j\), \(\epsilon_{ij}(t) = u_{i}(t) - u_{\text{rest}}\). 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_{\text{rest}} \end{equation}

SRM describes the membrane potential of neuron \(i\) as:

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

### Motivating Spiking Neural Networks

Since second-generation neural networks have excellent performance, why bother with spiking neural networks? In this section, we motivate spiking neural networks from various perspectives.

- Information Encoding

```
To directly compare ANNs and SNNs, one can consider the real-valued
outputs of ANNs to be the firing rate of a spiking neuron in steady
state. In fact, such rate coding has been used to explain
computational processes in the brain <a id="ffec32504761d0b966ab6c36eee992a6" href="#pfeiffer2018deep">(Pfeiffer \& Pfeil, 2018)</a>. Spiking
neuron models encode information beyond the average firing rate: these
models also utilize the relative timing between spikes
<a id="b77caeaea9e23a2bc9c87b5f10a91487" href="#guetig14_to_spike_or_when_to_spike">(Robert G\"utig, 2014)</a>, or spike phases (in-phase or
out-of-phase). These time-dependent codes are termed temporal codes,
and play an important role in biology. First, research has shown that
different actions are taken based on single spikes
<a id="b7885fc5161a6ad3286ee312ffcc0c0b" href="#stemmler96_singl_spike_suffic">(Martin Stemmler, 1996)</a>. Second, relying on the average firing rate
would greatly increase the latency of the brain, and our brain often
requires decision-making long before several spikes are accumulated.
It has also been successfully demonstrated that temporal coding
achieves competitive empirical performance on classification tasks for
both generated datasets, as well as image datasets like MNIST and
CIFAR <a id="caaddec51f6948e5fea79b6d41c79676" href="#comsa19_tempor_codin_spikin_neural_networ">(Comsa et al., 2019)</a>.
```

```
A faction of the machine learning and neurobiology community strives
for emulation of the biological brain. There are several
incompatibilities between ANNs and the current state of neurobiology
that are not easily reconciliated.
First, neurons in ANNs communicate via continuous-valued activations.
This is contrary to neurobiological research, which shows that
communication between biological neurons communicate by broadcasting
spike trains: trains of action potentials to downstream neurons. The
spikes are to a first-order approximation of uniform amplitude, unlike
the continuous-valued activations of ANNs.
Second, backpropagation as a learning procedure also presents
incompatibilities with the biological brain <a id="cf477ffda17db950a7fd58c9d833182b" href="#TAVANAEI201947">("Amirhossein Tavanaei et al., 2019)</a>.
Consider the chain rule in backpropagation:
\begin{equation} \label{chainrule}
\delta\_{j}^{\mu}=g^{\prime}\left(a\_{j}^{\mu}\right) \sum\_{k} w\_{k j} \delta\_{k}^{\mu}
\end{equation}
\\(\delta\_{j}^{\mu}\\) and \\(\delta\_{k}^{\mu}\\) denote the partial
derivatives of the cost function for input pattern \\(\mu\\) with respect
to the net input to some arbitrary unit \\(j\\) or \\(k\\). Unit \\(j\\) projects
feed-forward connections to the set of units indexed by \\(k\\).
\\(g(\cdot)\\) is the activation function applied to the net input of unit
\\(j\\), denoted \\(a\_j^{\mu}\\), \\(w\_{kj}\\) are the feedforward weights
projecting from unit \\(j\\) to the set of units indexed by \\(k\\).
The chain rule formulation presents two problems. First, the
gradients \\(g'(\cdot)\\) requires derivatives, but \\(g(\cdot)\\) in spiking
neurons is represented by sum of Dirac delta functions, for which
derivatives do not exist. Second, the expression \\(\sum\_{k} w\_{k j}
\delta\_{k}^{\mu}\\) uses feedforward weights in a feedback fashion. This
mean that backpropagation is only possible in the presence of
symmetric feedback weights, but these do not exist in the brain. In
addition, during backpropagation the error assignment for each neuron
is computed using non-local information.
```

```
In a traditional Von Neumann architecture, the logic core operates on
data fetched sequentially from memory. In contrast, in neuromorphic
chips both computation and memory are distributed across computational
units that are connected via synapses. The neuronal architecture and
parameters hence play a key role in information representation and
define the computations that are performed.
It has also been observed that spike-trains in the mammalian brain are
often sparse in time, suggesting that timing and relative timings of
spikes encode large amounts of information. Neuromorphic chips
implement this same sparse, low-precision communication protocol
between neurons on the chip, and by offering the same asynchronous,
event-based parallelism paradigm that the brain uses, are able to
perform certain workloads with much less power than Von Neumann chips.
These integrated circuits are typically programmed with spiking neural
networks. Examples of such chips include IBM's TrueNorth
<a id="ff5e58b063893165385318f053dc98c9" href="#Merolla668">(Merolla et al., 2014)</a> and Intel's Loihi <a id="8146155533911feae2a67ddf637b1e29" href="#davies2018loihi">(Davies et al., 2018)</a>. Because
spiking neural networks have not yet been successfully trained on many
tasks, neuromorphic chips has seen little practical use. These chips
have only recently been successfully used in robotic navigation
<a id="162706e1b73312c94623c3c6304f9856" href="#SnnSlam">("Tang et al., 2019)</a>, and solving graph problems by manual construction of the
network graph <a id="022dd7a783479dfb761bf88b30399ee3" href="#Severa2016SpikingNA">("William Severa et al., 2016)</a>.
```

### Training Spiking Neural Networks

As explained in

- Non-gradient based methods

```
Spiking neurons communicate via spikes, hence, unlike ANNs, gradients
are non-existent. In addition, backpropagation is not biologically
plausible (see <bioplausible>). This motivates the use of
plasticity-based methods and evolutionary strategies for training
SNNs.
One category of learning rules used in SNNs are local learning rules.
These rules include Hebbian learning (neurons that fire together wire
together), and its extension: the spike-timing-dependent-plasticity
rule (STDP). Inspired by experiments in neuroscience, central to these
learning rules is the theme that neuron spike ordering and their
relative timings encode information. STDP adjusts the strength of
connections between neurons using the relative timing of a neuron's
output and its input potentials (hence, spike-timing dependent).
In machine learning terminology, the weights of the synapses are
adjusted according to fixed rules for each training example. Each
synapse is given a weight \\(0 \le w \le w\_{max}\\) , characterizing its
strength, and its change depends on the exact moments \\(t\_{pre}\\) of
pre-synaptic spikes and \\(t\_{post}\\) of post-synaptic spikes
<a id="5c8bcfe50f375b189e564db4d78fe1a3" href="#sboev18_spikin_neural_networ_reinf_learn">(Alexander Sboev et al., 2018)</a>:
\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}
where \\(\tau\_{+}\\) and \\(\tau\_{-}\\) are time constants. \\(\tau\_{+} = 16.8ms\\)
and \\(\tau\_{-} = 33.7ms\\) are reasonable approximations obtained
experimentally.
There are several libraries like BindsNET
<a id="b266899abe9f6133f1e6d55ad24513db" href="#10.3389/fninf.2018.00089">(Hazan et al., 2018)</a> that simulate SNNs on Von Neumann
computers implementing these rules. Recent attempts have been made to
combine Reinforcement Learning and STDP: both in solving RL problems
<a id="b266899abe9f6133f1e6d55ad24513db" href="#10.3389/fninf.2018.00089">(Hazan et al., 2018)</a>, and using the reinforcement learning
framework to train SNN
<a id="54e69da24e936643028e5c2591a092e6" href="#10.3389/fnbot.2019.00018">(Bing et al., 2019)</a><a>, </a><a id="cb84873c6f16a361dfb01de00ae1425a" href="#10.3389/fnins.2018.00435">(Lee et al., 2018)</a>. However, SNNs
trained using the STDP learning rule have yet to achieve comparable
performance compared to ANNs on relatively simple datasets like MNIST
<a id="cf477ffda17db950a7fd58c9d833182b" href="#TAVANAEI201947">("Amirhossein Tavanaei et al., 2019)</a>.
```

- Gradient-based methods

```
Performance is important for practical applications, and
gradient-based training methods such as backpropagation has shown
competitive performance. It is thus desirable to train spiking neural
networks with these gradient-based methods.
There are several problems with spike-compatible gradient-based
methods. First, most of these methods cannot train neurons in the
hidden layers: they can only train neurons at the final layer, that
receive the desired target output pattern
<a id="78cca07dd56a8c50ebd16d663c3b095d" href="#urbanczik09_gradien_learn_rule_tempot">(Robert Urbanczik \& Walter Senn, 2009)</a><a>, </a><a id="9a5e9b63107e695a46943df33a811909" href="#training_deep_snn_bpp_lee">(Lee et al., 2016)</a>.
Second, the discontinuous, binary nature of spiking output needs to be
addressed. For example, SpikeProp approximates the membrane
threshold function at a local area with a linear function, introducing
gradients and computing the exact formulae for error backpropagation
for synaptic weights and spike times <a id="41ed49a4572e3909921301127000035c" href="#spikeprop">(Bohte et al., 2000)</a>. Others have
modified the threshold function with a gate function
<a id="2b991fd4ef5412f89154985cec53e4e8" href="#NIPS2018_7417">(Huh \& Sejnowski, 2018)</a>, used the alpha transfer function to derive
gradient update rules <a id="caaddec51f6948e5fea79b6d41c79676" href="#comsa19_tempor_codin_spikin_neural_networ">(Comsa et al., 2019)</a>,
and approximate the dirac-delta spikes with a probability density
function <a id="123af3c690cf296b196d898878568449" href="#NIPS2018_7415">(Shrestha \& Orchard, 2018)</a>.
Another approach is converting trained ANN models into SNNs
<a id="f842b3cf76d4cc279ae01a4cf75c48ba" href="#rueckauer16_theor_tools_conver_analog_to">(Rueckauer et al., 2016)</a>. Common ANN layers such
as softmax, batch normalization and max-pooling layers have their
corresponding spiking counterparts.
Equilibrium Propagation was recently proposed to solve the
neurobiological incompatibilities of backpropagation
<a id="38c93c265031c562bcf3ab3a89b9d896" href="#10.3389/fncom.2017.00024">(Scellier \& Bengio, 2017)</a>. Because the gradients are defined only
in terms of local perturbations, the synaptic updates correspond to
the standard form of STDP. The propagated signal encodes the gradients
of a well-defined objective function on energy-based models, where the
goal is to minimize the energy of the model. To resolve the issue of
communication using binary-valued signals, step-size annealing was
used to train spiking neural networks with Equilibrium Propagation
<a id="d1886fb7b4cde1636a730b6072eb45c5" href="#pmlr-v89-o-connor19a">(O'Connor et al., 2019)</a>.
```

- Future Research Areas

```
A nascent area is local learning on neuromorphic chips. Thus far
spiking neural networks are simulated and trained before deployment on
a neuromorphic chip. In Intel's Loihi chip, each core contains a
learning engine that can update synaptic weights using the 4-bit
microcode-programmed learning rules that are associated with that
synapse. This opens up areas for online learning.
```

## 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.

## TODO Surrogate Gradient Learning in Spiking Neural Networks (Neftci et al., 2019)

## TODO Theories of Error Back-Propagation in the Brain (James Whittington & Rafal Bogacz, 2019)

## §comsa2019_temp_coding

## 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.

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