Exponential integrate-and-fire

In biology exponential integrate-and-fire models are compact and computationally efficient nonlinear spiking neuron models with one or two variables. The exponential integrate-and-fire model was first proposed as a one-dimensional model.^[1] The most prominent two-dimensional examples are the adaptive exponential integrate-and-fire model^[2] and the generalized exponential integrate-and-fire model.^[3] Exponential integrate-and-fire models are widely used in the field of computational neuroscience and spiking neural networks because of (i) a solid grounding of the neuron model in the field of experimental neuroscience, (ii) computational efficiency in simulations and hardware implementations, and (iii) mathematical transparency.

Exponential integrate-and-fire (EIF)

The exponential integrate-and-fire model (EIF) is a biological neuron model, a simple modification of the classical leaky integrate-and-fire model describing how neurons produce action potentials. In the EIF, the threshold for spike initiation is replaced by a depolarizing non-linearity. The model was first introduced by Nicolas Fourcaud-Trocmé, David Hansel, Carl van Vreeswijk and Nicolas Brunel.^[1] The exponential nonlinearity was later confirmed by Badel et al.^[4] It is one of the prominent examples of a precise theoretical prediction in computational neuroscience that was later confirmed by experimental neuroscience.

In the exponential integrate-and-fire model,^[1] spike generation is exponential, following the equation:

${\displaystyle {\frac {dV}{dt}}-{\frac {R}{\tau _{m}}}I(t)={\frac {1}{\tau _{m}}}[E_{m}-V+\Delta _{T}\exp \left({\frac {V-V_{T}}{\Delta _{T}}}\right)]}$.

Parameters of the exponential integrate-and-fire neuron can be extracted from experimental data.^[4]

where ${\displaystyle V}$ is the membrane potential, ${\displaystyle V_{T}}$ is the intrinsic membrane potential threshold, ${\displaystyle \tau _{m}}$ is the membrane time constant, ${\displaystyle E_{m}}$ is the resting potential, and ${\displaystyle \Delta _{T}}$ is the sharpness of action potential initiation, usually around 1 mV for cortical pyramidal neurons.^[4] Once the membrane potential crosses ${\displaystyle V_{T}}$, it diverges to infinity in finite time.^[5][4] In numerical simulation the integration is stopped if the membrane potential hits an arbitrary threshold (much larger than ${\displaystyle V_{T}}$) at which the membrane potential is reset to a value V_r . The voltage reset value V_r is one of the important parameters of the model.

Two important remarks: (i) The right-hand side of the above equation contains a nonlinearity that can be directly extracted from experimental data.^[4] In this sense the exponential nonlinearity is not an arbitrary choice but directly supported by experimental evidence. (ii) Even though it is a nonlinear model, it is simple enough to calculate the firing rate for constant input, and the linear response to fluctuations, even in the presence of input noise.^[6]

A didactive review of the exponential integrate-and-fire model (including fit to experimental data and relation to the Hodgkin-Huxley model) can be found in Chapter 5.2 of the textbook Neuronal Dynamics.^[7]

Adaptive exponential integrate-and-fire (AdEx)

Initial bursting AdEx model

The adaptive exponential integrate-and-fire neuron ^[2] (AdEx) is a two-dimensional spiking neuron model where the above exponential nonlinearity of the voltage equation is combined with an adaptation variable w

${\displaystyle \tau _{m}{\frac {dV}{dt}}=RI(t)+[E_{m}-V+\Delta _{T}\exp \left({\frac {V-V_{T}}{\Delta _{T}}}\right)]-Rw}$

${\displaystyle \tau {\frac {dw(t)}{dt}}=-a[V_{\mathrm {m} }(t)-E_{\mathrm {m} }]-w+b\tau \delta (t-t^{f})}$

where w denotes an adaptation current with time scale ${\displaystyle \tau }$. Important model parameters are the voltage reset value V_r, the intrinsic threshold ${\displaystyle V_{T}}$, the time constants ${\displaystyle \tau }$ and ${\displaystyle \tau _{m}}$ as well as the coupling parameters a and b. The adaptive exponential integrate-and-fire model inherits the experimentally derived voltage nonlinearity ^[4] of the exponential integrate-and-fire model. But going beyond this model, it can also account for a variety of neuronal firing patterns in response to constant stimulation, including adaptation, bursting and initial bursting.^[8]

The adaptive exponential integrate-and-fire model is remarkable for three aspects: (i) its simplicity since it contains only two coupled variables; (ii) its foundation in experimental data since the nonlinearity of the voltage equation is extracted from experiments;^[4] and (iii) the broad spectrum of single-neuron firing patterns that can be described by an appropriate choice of AdEx model parameters.^[8] In particular, the AdEx reproduces the following firing patterns in response to a step current input: neuronal adaptation, regular bursting, initial bursting, irregular firing, regular firing.^[8]

A didactic review of the adaptive exponential integrate-and-fire model (including examples of single-neuron firing patterns) can be found in Chapter 6.1 of the textbook Neuronal Dynamics.^[7]

Generalized exponential integrate-and-fire Model (GEM)

The generalized exponential integrate-and-fire model^[3] (GEM) is a two-dimensional spiking neuron model where the exponential nonlinearity of the voltage equation is combined with a subthreshold variable x

${\displaystyle \tau _{m}{\frac {dV}{dt}}=RI(t)+[E_{m}-V+\Delta _{T}\exp \left({\frac {V-V_{T}}{\Delta _{T}}}\right)]-b\,[E_{x}-V]x}$

${\displaystyle \tau _{x}(V){\frac {dx(t)}{dt}}=x_{0}(V_{\mathrm {m} }(t))-x}$

where b is a coupling parameter, ${\displaystyle \tau _{x}(V)}$ is a voltage-dependent time constant, and ${\displaystyle x_{0}(V)}$ is a saturating nonlinearity, similar to the gating variable m of the Hodgkin-Huxley model. The term ${\displaystyle b[E_{x}-V]x}$ in the first equation can be considered as a slow voltage-activated ion current.^[3]

The GEM is remarkable for two aspects: (i) the nonlinearity of the voltage equation is extracted from experiments;^[4] and (ii) the GEM is simple enough to enable a mathematical analysis of the stationary firing-rate and the linear response even in the presence of noisy input.^[3]

A review of the computational properties of the GEM and its relation to other spiking neuron models can be found in.^[9]