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The Quadratic Integrate-and-Fire Neuron

The Hodgkin–Huxley model can be simplified into various 1-D systems (and 2-D and 3-D, as it is originally a 4-D system). Even in the 1-D case, we can identify four fundamental models: one for each combination of activation vs. inactivation gating and inward vs. outward current.

The typical choice for preliminary analysis is the so-called Leak + Instantaneous Sodium Model, often referred to as the persistent sodium model. In this model, we consider \(p\) as a single activation gate variable (as opposed to the 4-D Hodgkin–Huxley model with three gating variables).

Essentially, this captures the voltage dynamics for a neuron with only one type of channel (sodium channel) and with only one activation gate. The model also has a leaky outward current (typically representing potassium \(K^+\)). The sodium current here is persistent, meaning that once activated, it has no inactivation mechanism— it continues to flow indefinitely.

\[ C\dot{V} = I - g_L(V-E_L) - gp(V-E) \]
\[ \dot{p} = \frac{(p_{\infty}(V) - p)}{\tau(V)} \]

To simplify, we assume the gating variable \(p\) responds much faster than voltage dynamics, and thus treat it as instantaneous:

\[ p = p_{\infty}(V) \]

Substituting:

\[ C\dot{V} = I - g_L(V-E_L) - g p_{\infty}(V)(V-E) \]

Considering sodium specifically, we write:

\[ C\dot{V} = I - g_L(V-E_L) - g_{Na} m_{\infty}(V)(V-E_{Na}) \]

With

\[ m_{\infty}(V) = \frac{1}{1+e^{\frac{V_{1/2}-V}{k}}} \]
\[ C = 10\,\mu F,\quad I = 0\ \text{pA},\quad g_L = 19\, mS,\quad E_L = -67\ \text{mV} \]
\[ g_{Na} = 74\, mS,\quad V_{1/2} = 1.5\, mV,\quad k = 16\, mV,\quad E_{Na} = 60\, mV \]

Substituting numerical values:

\[ 10\dot{V} = -19(V+67) - 74\left(\frac{1}{1+e^{\frac{1.5-V}{16}}}\right)(V-60) \]

No analytic solution exists for this nonlinear ODE, so we proceed with qualitative analysis.

Fixed Point Analysis

Fixed points satisfy \(\dot{V} = 0\):

\[ -19(V+67) - 74\left(\frac{1}{1+e^{\frac{1.5-V}{16}}}\right)(V-60) = 0 \]

Numerically, this yields three fixed points: \(x_1 \approx -52.51,\; x_2 \approx -41.51,\; x_3 \approx 30.95\).

sodium

The first and third are stable equilibria (rest and excited states), separated by the unstable threshold \(x_2\).

phase portrait

Bifurcation Analysis

Varying the input current \(I\) alters stability. At \(I=16\), the two leftmost fixed points merge and annihilate in a saddle-node bifurcation.

bifurcation diagram

Hartman–Grobman Theorem

Locally, near a hyperbolic fixed point \(V_{eq}\), the system

\[ \dot{V} = F(V) \]

is topologically equivalent to

\[ \dot{V} = \lambda (V-V_{eq}), \qquad \lambda = F'(V_{eq}) \]

Topological Normal Form

For a saddle-node bifurcation, the normal form is:

\[ \dot{V} = I + V^2 \]

This is the Quadratic Integrate-and-Fire Neuron.

Simulations

Numerical experiments reveal regimes depending on \(I\):

I < 0

Two equilibria at \(\pm\sqrt{|I|}\). Resetting just above threshold induces tonic spiking:

non-tonic reset equilibrium tonic spiking

I = 0

Single equilibrium at \(0\). Dynamics depend on reset value:

initial spike tonic spiking equilibrium reset

I > 0

No equilibria. All trajectories spike repetitively:

fast spiking slow spiking

These cover the main dynamical properties of the Quadratic Integrate-and-Fire Neuron.