= Leaky integrate-and-fire model
{title2=$C\dot V=-g_L(V-E_L)+I(t)$}
= LIF model
{c}
{synonym}
The <leaky integrate-and-fire model> combines a passive equation for <membrane potential> with a firing rule. Between events, inward-positive applied <electric current> $I(t)$ obeys
$$
C\frac{dV}{dt}=-g_L(V-E_L)+I(t),\qquad R=\frac{1}{g_L},\qquad \tau=\frac{C}{g_L}=RC,
$$
where $C>0$ is membrane <capacitance>, $g_L>0$ is the leak conductance, $R$ is the corresponding <electrical resistance>, and $E_L$ is the leak <reversal potential>. Here the leak conductance is constant. The <time constant> $\tau$ describes passive relaxation. When $V$ first reaches a threshold $V_\theta$ from below, record an <action potential> event and reset $V$ to $V_r<V_\theta$. In the version used here, hold $V=V_r$ for a fixed <neuronal refractory period> $t_{\mathrm{ref}}$, then resume integration. See <Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition by Wulfram Gerstner and others/1.3 Integrate-And-Fire Models>.
For constant $I$, with $t=0$ at the start of an integration interval and $V(0)=V_{\mathrm{initial}}$, the exact subthreshold solution is
$$
V(t)=E_L+RI+(V_{\mathrm{initial}}-E_L-RI)e^{-t/\tau}.
$$
The <equilibrium point> of this passive equation is $V_\infty=E_L+RI$. Subtracting $V_\infty$ leaves $d(V-V_\infty)/dt=-(V-V_\infty)/\tau$, explaining the <exponential function>. This solution applies only until a threshold event or a change in $I$. Starting below threshold, constant input produces repeated events if $V_\infty>V_\theta$; if $V_\infty=V_\theta$, the threshold is approached asymptotically and never reached at finite time.
For a worked example, choose
$$
E_L=V_r=V_{\mathrm{initial}}=-65\,\mathrm{mV},\quad V_\theta=-50\,\mathrm{mV},\quad \tau=20\,\mathrm{ms},\quad R=100\,\mathrm{M\Omega},\quad t_{\mathrm{ref}}=2\,\mathrm{ms}.
$$
These imply $g_L=10\,\mathrm{nS}$ and $C=200\,\mathrm{pF}$. All these parameters are illustrative model choices, not measurements from a particular <neuron>. With $I=200\,\mathrm{pA}$, <Ohm's law> gives $RI=20\,\mathrm{mV}$ and hence $V_\infty=-45\,\mathrm{mV}$. Thus
$$
V(t)=-45\,\mathrm{mV}-20\,\mathrm{mV}\,e^{-t/(20\,\mathrm{ms})}.
$$
At threshold, $e^{-t/\tau}=1/4$, so the first event occurs after $T=20\ln4\,\mathrm{ms}\approx27.7259\,\mathrm{ms}$. Each subsequent cycle includes the $2\,\mathrm{ms}$ <neuronal refractory period>, giving
$$
f=\frac{1}{T+t_{\mathrm{ref}}}\approx33.6407\,\mathrm{Hz}.
$$
This is the ideal sustained firing rate; the first-event latency is $T$, not $T+t_{\mathrm{ref}}$.
With $I=100\,\mathrm{pA}$ instead, $RI=10\,\mathrm{mV}$ and $V_\infty=-55\,\mathrm{mV}$. From the same initial state, $V(t)=-55\,\mathrm{mV}-10\,\mathrm{mV}\,e^{-t/\tau}$ stays below $-50\,\mathrm{mV}$: no events occur, however long this current is held. The critical sustained current is $(V_\theta-E_L)/R=150\,\mathrm{pA}$; equality gives asymptotic approach rather than finite-time firing.
\Image[/integrate-and-fire-current-steps.png]
{title=Exact leaky integrate-and-fire responses to two current steps}
{description=Both inputs start at $20\,\mathrm{ms}$ and end at $180\,\mathrm{ms}$. A $100\,\mathrm{pA}$ step approaches $-55\,\mathrm{mV}$ without firing; a $200\,\mathrm{pA}$ step produces threshold events separated by $29.7259\,\mathrm{ms}$ after the first. Orange ticks mark events, and shaded intervals mark the $2\,\mathrm{ms}$ reset hold. The dashed $-45\,\mathrm{mV}$ level is the stronger input's passive equilibrium, which the reset trajectory never reaches. These are analytic model trajectories, not recordings or biological spike waveforms.}
The leak gives old inputs progressively less influence, while <capacitance> prevents an ordinary finite <electric current> step from changing $V$ instantly. This combination makes the <leaky integrate-and-fire model> useful for exploring input summation, event timing and network activity at low computational cost. Its fixed parameters and reset omit the evolving <voltage-gated ion channels> of the <Hodgkin-Huxley model>, spatial integration in <dendrites> and mechanisms of <neural adaptation>. Treating a <synapse> as a prescribed $I(t)$ also omits its voltage-dependent driving force unless a conductance term is added. See <Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition by Wulfram Gerstner and others/1.4 Limitations of the Leaky Integrate-and-Fire Model> and <constant-input firing rate of a leaky integrate-and-fire neuron>.
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