1.1.2 Point Neuron Models

Point Neurons Models are minimalistic spiking neuron models that use simplified electro-mathematical equations to mimic the spike generation, voltage reset, and refractory state behaviours of biological neurons; they conveniently ignore to model the ionic channels, axial conductance, axonal propagation of spike/action-potential, etc., basically anything that relates to the spatial form of a biological neuron. Common examples of point spiking neurons are Integrate & Fire, Leaky Integrate & Fire, and Resonate & Fire spiking neuron models. We explain each of these three point neuron models below, along with their code examples. We highly encourage you to play with their different hyper-parameter settings, and observe/analyse their spiking behaviour.

Notations:

Before we explain the various spiking models of point neurons, we first introduce the mathematical notations (of their variables) that we henceforth use; these notations are consistent in their meaning throughout the book, unless stated otherwise.

• Spike: The incoming/outgoing spike to/from a spiking neuron is denoted as $S(t)$ in continuous time or $S[t]$ in discrete time.

• Current: The input current (to a spiking neuron) is denoted as $I(t)$ in continuous time or$I[t]$ in discrete time.

• Voltage: The voltage/membrane potential of a spiking neuron is denoted as $V(t)$ in continuous time or $V[t]$ in discrete time.

1.1.2.1Integrate & Fire Neuron Model¶

Integrate & Fire ($\texttt{IF}$) neuron model is the simplest of all the spiking neuron models -- by virtue of which, it’s dynamic behaviour is also quite limited. An $\texttt{IF}$ neuron, as the name goes, integrates the incoming/input spikes into its non-decaying membrane potential/voltage, followed by generating an output spike when its voltage reaches/crosses the set voltage threshold; whether or not it undergoes through a refractory state, is dependent on its subjective implementation. Following is the continuous-time equation of an $\texttt{IF}$ neuron Heeger, 2000 :

where, $C_\text{m}$ is the neuron’s Capacitance value (generally assumed to be 1), $V(t)$ is neuron’s membrane potential/voltage, and $I(t)$ is the input stimulus/current due to the incoming spikes.

In Eq (1), note the inherent temporality in $V(t)$ and $I(t)$ by the virtue of them being a function of time $t$. Here, we subtly note the activation function of an example artificial neuron, say $\texttt{ReLU}$: $max(x, 0)$, where $x\in\mathbb{R}$; and as it can be seen in the $\texttt{ReLU}$ neuron, there is no temporal component in it! This is one of the stark differences between between spiking neurons and artificial neurons, i.e.,

Other few differences are:

• Unlike artificial neurons, spiking neurons are (implicitly) recurrent (note the relation between $V[t]$ and $V[t-1]$).

• Spiking neurons output discrete spikes (which can be graded too, i.e., non-binary integers).

1.1.2.1.1Implementing $\texttt{IF}$ neuron¶

Coming back to the $\texttt{IF}$ neurons, to implement it on a digital system, i.e., a computer, it is necessary to discretize its continuous-time equation such that one can program it. We can discretize the Eq (1) using the forward Euler method -- we show how to do it in the Discretizing $\texttt{IF}$ neuron. For quick reference, following are the discrete-time equations that describe an $\texttt{IF}$ neuron:

Before we begin explaining the Eqs (2), let us introduce the concept of Spike Train.

A spike train holding a random collection of binary spikes and feeding them to an $\texttt{IF}$ neuron (see Figure 1) is shown below.

In Eq (2)$\textsf{a}$, $I[t]$ and $I[t-1]$ are the $\texttt{IF}$ neuron’s current at time-step $t$ and $t-1$, $i_\text{decay}$ is the current decay value (such that, $0\leq i_\text{decay}\leq1$). As can be seen in the current update Eq (2)$\textsf{a}$, the current at time-step $t$ i.e., $I[t]$ is a sum of the decayed value of current at the previous time-step $t-1$ i.e., $I[t-1]$ (decayed by the factor $(1 - i_\text{decay})$) and the $w$ weighted spike input $S_\text{inp}[t]$. Note that the $S_\text{inp}[t]$ can be from a pre-synaptic neuron too (instead of just being a part of the above randomly generated spike train).

In Eq (2)$\textsf{b}$, $V[t]$ and $V[t−1]$ are the $\texttt{IF}$ neuron’s voltage at the time-steps $t$ and $t-1$. Note that unlike Eq (2)$\textsf{a}$, no decayed value of $V[t-1]$ is added to $I[t]$ to obtain the updated $V[t]$.

In Eq (2)$\textsf{c}$ $V_\text{thr}$ is the threshold voltage and $\Theta$ is the Heaviside step-function that outputs 1 for positive arguments and 0 for non-positive arguments. As can be seen in the spike output Eq (2)$\textsf{c}$, a binary spike $S_\text{out}[t]$ (which can either be a 0 or a 1) is generated by the $\texttt{IF}$ neuron at the time-step $t$ if it’s voltage $V[t]$ crosses the $V_\text{thr}$. Note that $V_\text{thr}$ here is fixed, however, a few implementations also keep it variable/adaptable over time.

In the Eq (2)$\textsf{d}$, $V_\text{rest}$ is the resting membrane potential/voltage value, which is generally set as 0. As explained before, the two ways to hard reset and soft reset are shown in the voltage reset Eq (2)$\textsf{d}$, either of which can be chosen for the implementation. The operational difference between these two implementations of voltage reset will be clear when we will implement them in code later. For now, a hard reset sets $V[t]$ to $V_\text{rest}$ after the neuron produces a spike and a soft reset instead subtracts $V_\text{thr}$ from $V[t]$. In other words, the soft reset enables the neuron to retain some contribution of the $I[t]$ or $S_\text{inp}[t]$ received, while hard reset results the neuron to discard any such contribution and start updating its $V[t]$ right from $V_\text{rest}$.

1.1.2.2Leaky Integrate & Fire Neuron Model¶

Leaky Integrate & Fire ($\texttt{LIF}$) neuron model is another simple and a common spiking neuron model that displays slightly more complex behavior than the $\texttt{IF}$ neuron model. A $\texttt{LIF}$ neuron, as the name goes, integrates the incoming/input spikes into its decaying membrane potential/voltage, followed by generating an output spike when its voltage reaches/crosses the set voltage threshold. Note that the refractory period (after spiking) is a standard component of the $\texttt{LIF}$ neuron model and is often included in its implementation, although, one can choose to ignore this. Since the refractory period in a spiking neuron prevents it from spiking immediately after it has spiked, it influences the $\texttt{LIF}$ neuron’s firing (rate) behavior i.e., its response to the input stimuli. Following is the continuous-time equation (Eq (3)) of the $\texttt{LIF}$ neuron Gerstner et al., 2014:

where, $\tau_\text{m}$ and $R_\text{m}$ are the neuron’s time-constant and membrane resistance respectively. Note $\tau_\text{m}=R_\text{m}C_\text{m}$ and $R_\text{m}$ contributes to the “leak” of accumulating charge in the membrane capacitance $C_\text{m}$, hence the name: “leaky integrator” Gerstner et al., 2014.

1.1.2.2.1Implementing $\texttt{LIF}$ neuron¶

Similar to implementing the $\texttt{IF}$ neuron on a digital computer, one can discretize the Eq (3) of the $\texttt{LIF}$ neuron via the forward Euler method; we do that in the Discretizing $\texttt{LIF}$ neuron. Following are the discrete-time equations that describe a $\texttt{LIF}$ neuron:

In Eq (4)$\textsf{a}$, $i_\text{decay}$ is the current decay value – same as defined for the $\texttt{IF}$ neuron; and in Eq (4)$\textsf{b}$, $v_\text{decay}$ is the voltage decay value (such that, $0 < v_\text{decay} < 1$).

Note that Eq (4)$\textsf{b}$ implements a decayed voltage accumulation - this is unlike the Eq (2)$\textsf{b}$, where voltage was accumulated without decay. However, if $v_\text{decay}=0$, i.e., no voltage decay, then the $\texttt{LIF}$ neuron becomes equivalent to the $\texttt{IF}$ neuron; and if $v_\text{decay}=1$, then the $\texttt{LIF}$ neuron does not account for its previous time-step’s (i.e., of $t-1$) voltage value. Thus, in a $\texttt{LIF}$ neuron, $v_\text{decay}$ cannot be a 0 or a 1. Rest of the Eqs (4)$\textsf{c}$ and (4)$\textsf{d}$ remain the same as described for the $\texttt{IF}$ neuron.

Henceforth, in all the chapters, $\texttt{IF}$ will imply Integrate & Fire spiking neuron (implemented with Eqs (2)) and $\texttt{LIF}$ will imply Leaky Integrate & Fire spiking neuron (implemented with Eqs (4)); where hard reset is chosen (for both $\texttt{IF}$ and $\texttt{LIF}$ neuron models) unless stated otherwise.

1.1.2.3Resonate & Fire neuron model¶

@incollection{snnbook2026-point-neurons,
  author    = {Gaurav, Ramashish and Pedersen, Jens Egholm and Bogdan, Petrut},
  title     = {{Point Neuron Models}},
  booktitle = {{Practical Spiking Neural Networks}},
  publisher = {Open Neuromorphic},
  year      = {2026},
  edition   = {Version 0.8},
  url       = {https://snnbook.net/topics/1_2_1_point-neurons},
}