Point Neurons Discretization

Here, we present the derivations based on the Forward Euler method to discretize the continuous time equations of the $\texttt{IF}$ and $\texttt{LIF}$ neurons. The Forward Euler method is a simple numerical method to approximate solutions to the first order Ordinary Differential Equations (ODEs). Note that the equations of the $\texttt{IF}$ (Eq (1)) and $\texttt{LIF}$ (Eq (3)) neurons are first order ODEs. Therefore, we can use the Forward Euler method to discretize and solve them numerically on our computers.

Forward Euler method¶

Consider

and an initial condition $y(t=0) = y_{0}$. In the Forward Euler method, a first order ODE is approximated (based on the forward difference derivative) as below:

where $h$ is a very small value, i.e., a very small change in time $t$. Let us sample $y(t)$ (defined over continuous time $t$) at discrete times $t_{n}$, i.e., $y[t_{n}]$. In the spirit of this discretization, let us also assume that the difference between consecutive discrete time-steps (i.e., $t_{n+1}$ and $t_{n}$) is very small, i.e., let $h = t_{n+1} - t_{n}$. Thus, the Eq (2) becomes the following:

For convenience, let us replace $y[t_{n+1}]$ and $y[t_{n}]$ with $y[n+1]$ and $y[n]$ respectively. Also, for practical purposes, let us replace ‘$\approx$’ with ‘$=$’, however, we should keep in mind that Forward Euler method is still an approximation and error grows with the value of $h$ as well as accumulates with every next time-step. Finally, rearranging the Eq (3) gives us the following:

We can use the formulation of the Eqs (4) or (3) to derive the discrete-time equations of the $\texttt{IF}$ and $\texttt{LIF}$ neurons.

Discretizing $\texttt{IF}$ neuron¶

The continuous-time equation of the $\texttt{IF}$ neuron as mentioned in the Section 1.1.2.1 is:

To discretize it, let us consider $C_\text{m}=1$, therefore, the Eq Equation to discretize becomes:

As can be easily seen, the Eq Equation is of the same form as Eq (1) above, thus, from Eq (4), we get the following discrete form of Eq Equation:

i.e.,

Note that in Eq Equation, $h$ denotes the difference between consecutive timesteps (as mentioned in Forward Euler method), therefore, let us assume the difference is 1 timestep, i.e., $h=1$. Thus, setting $h=1$ in Eq Equation, we get the following:

which is a discrete-time equation of an $\texttt{IF}$ neuron.

Discretizing $\texttt{LIF}$ neuron¶

The continuous-time equation of the $\texttt{LIF}$ neuron as mentioned in the Section 1.1.2.2 is:

which upon rearranging becomes:

Note that $\tau_\text{m}=R_\text{m}C_\text{m}$, therefore, Eq Equation becomes:

To discretize it, let us consider $C_\text{m}=1$ (same as for discretizing the $\texttt{IF}$ neuron), and $V_\text{rest}=0$, therefore, the Eq Equation to discretize becomes:

As can be easily seen, Eq Equation is of the same form as Eq (1), thus, from Eq (4), we get the following discrete form of Eq Equation:

i.e.,

Note that in Eq Equation (similar to the case of discretizing the $\texttt{IF}$ neuron), $h$ denotes the difference between consecutive timesteps, therefore, assuming $h=1$ in Eq Equation, we get the following:

which is the discrete-time equation of a $\texttt{LIF}$ neuron.

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