Neuron Models

The first "machine learning" book I read when I was in college was called How to Build a Brain by Chris Eliasmith. I've had cause to revisit biological neuron models recently and figured I would resurrect some old code I wrote for simulating neuron models.

Leaky Integrate-and-Fire

The Leaky Integrate-and-Fire (LIF) model is the simplest biologically-plausible neuron model. It treats the neuron as a leaky capacitor that integrates input current over time. When the membrane potential reaches a threshold, the neuron "fires" and resets.

The dynamics are governed by:

\tau \frac{dV}{dt} = -(V - V_{\text{rest}}) + RI

where $V$ is the membrane potential, $\tau$ is the membrane time constant, $V_{\text{rest}}$ is the resting potential, $R$ is the input resistance, and $I$ is the input current.

When $V \geq V_{\text{thresh}}$ , the neuron fires and $V$ is reset to $V_{\text{reset}}$ .

Despite its simplicity, the LIF model captures essential features of neuronal excitability and is widely used in large-scale neural simulations due to its computational efficiency.

Izhikevich

The Izhikevich model is a two-dimensional model that combines computational efficiency with the ability to reproduce a wide variety of neuronal firing patterns. It was designed to be as simple as the LIF model but as rich as Hodgkin-Huxley.

The model equations are:

\begin{align} \frac{dv}{dt} &= 0.04v^2 + 5v + 140 - u + I \\ \frac{du}{dt} &= a(bv - u) \end{align}

with the spike-reset condition: if $v \geq 30$ mV, then $v \leftarrow c$ and $u \leftarrow u + d$ .

Here $v$ is the membrane potential, $u$ is a recovery variable, and the parameters $a, b, c, d$ control the neuron's dynamics. By adjusting these parameters, the model can reproduce over 20 different types of cortical neurons, including regular spiking, intrinsic bursting, chattering, and fast-spiking behaviors.

FitzHugh-Nagumo

The FitzHugh-Nagumo model is a simplified two-dimensional reduction of the Hodgkin-Huxley model, designed to capture the qualitative behavior of excitable systems while being amenable to mathematical analysis. It was independently developed by Richard FitzHugh and J. Nagumo in the 1960s.

The model is described by:

\begin{align} \frac{dv}{dt} &= v - \frac{v^3}{3} - w + I \\ \frac{dw}{dt} &= \frac{1}{\tau}(v + a - bw) \end{align}

where $v$ represents the membrane potential (fast variable), $w$ is a recovery variable (slow variable), and $I$ is the input current. The cubic nonlinearity in the first equation creates the excitability threshold.

This model is particularly useful for understanding the phase-plane dynamics of excitable systems and has been widely applied in studying oscillations, wave propagation, and pattern formation in biological systems.

Morris-Lecar

The Morris-Lecar model was developed in 1981 to describe the voltage oscillations in barnacle muscle fibers. It's a two-dimensional conductance-based model that includes calcium and potassium currents, making it biophysically more realistic than FitzHugh-Nagumo while remaining mathematically tractable.

The model equations are:

\begin{align} C_m \frac{dV}{dt} &= I - g_{Ca} m_\infty(V)(V - E_{Ca}) - g_K w(V - E_K) - g_L(V - E_L) \\ \frac{dw}{dt} &= \phi \frac{w_\infty(V) - w}{\tau_w(V)} \end{align}

where:

\begin{align} m_\infty(V) &= \frac{1}{2}\left(1 + \tanh\left(\frac{V - V_1}{V_2}\right)\right) \\ w_\infty(V) &= \frac{1}{2}\left(1 + \tanh\left(\frac{V - V_3}{V_4}\right)\right) \\ \tau_w(V) &= \frac{1}{\cosh\left(\frac{V - V_3}{2V_4}\right)} \end{align}

Here $V$ is the membrane potential, $w$ is the potassium channel activation, and $m_\infty$ represents the instantaneous calcium channel activation. The model can exhibit both Class I (continuous frequency-current relationship) and Class II (discontinuous) excitability depending on parameter choices.

Hodgkin-Huxley

The Hodgkin-Huxley model is the foundational conductance-based model of neuronal action potentials, developed by Alan Hodgkin and Andrew Huxley in 1952 based on their voltage-clamp experiments on the giant squid axon. This work earned them the Nobel Prize in Physiology or Medicine in 1963.

The model describes the dynamics of voltage-gated sodium and potassium channels:

C_m \frac{dV}{dt} = I - g_{Na} m^3 h (V - E_{Na}) - g_K n^4 (V - E_K) - g_L (V - E_L)

where the gating variables $m, h, n$ follow first-order kinetics:

\frac{dx}{dt} = \alpha_x(V)(1-x) - \beta_x(V)x \quad \text{for } x \in \{m, h, n\}

The rate functions $\alpha$ and $\beta$ are voltage-dependent and were empirically determined by Hodgkin and Huxley. Here:

$m$ represents sodium activation (fast)
$h$ represents sodium inactivation (slower)
$n$ represents potassium activation (slow)

The Hodgkin-Huxley model remains the gold standard for biophysically detailed neuron modeling and has been extended to include dozens of additional ion channel types in modern computational neuroscience.