exponential integrate-and-fire

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.{{Cite journal|last1=Fourcaud-Trocmé|first1=Nicolas|last2=Hansel|first2=David|last3=van Vreeswijk|first3=Carl|last4=Brunel|first4=Nicolas|date=2003-12-17|title=How Spike Generation Mechanisms Determine the Neuronal Response to Fluctuating Inputs|url= |journal=The Journal of Neuroscience|language=en|volume=23|issue=37|pages=11628–11640|doi=10.1523/JNEUROSCI.23-37-11628.2003|issn=0270-6474|pmc=6740955|pmid=14684865}} The exponential nonlinearity was later confirmed by Badel et al. 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, spike generation is exponential, following the equation:

: $\backslash frac\{dV\}\{dt\}\; -\; \backslash frac\{R\}\; \{\backslash tau\_m\}\; I(t)=\; \backslash frac\{1\}\; \{\backslash tau\_m\}[E\_m-V+\backslash Delta\_T\; \backslash exp\; \backslash left(\; \backslash frac\{V\; -\; V\_T\}\; \{\backslash Delta\_T\}\; \backslash right)]$.

File:Exponential Integrate-and-Fire (schematic).jpg

where $V$ is the membrane potential, $V\_T$ is the intrinsic membrane potential threshold, $\backslash tau\_m$ is the membrane time constant, $E\_m$ is the resting potential, and $\backslash Delta\_T$ is the sharpness of action potential initiation, usually around 1 mV for cortical pyramidal neurons. Once the membrane potential crosses $V\_T$, it diverges to infinity in finite time.{{cite journal|vauthors=Ostojic S, Brunel N, Hakim V|date=August 2009|title=How connectivity, background activity, and synaptic properties shape the cross-correlation between spike trains|journal=The Journal of Neuroscience|volume=29|issue=33|pages=10234–53|doi=10.1523/JNEUROSCI.1275-09.2009|pmc=6665800|pmid=19692598}}{{cite journal|authorlink5=Wulfram Gerstner|vauthors=Badel L, Lefort S, Brette R, Petersen CC, Gerstner W, Richardson MJ|date=February 2008|title=Dynamic I-V curves are reliable predictors of naturalistic pyramidal-neuron voltage traces|journal=Journal of Neurophysiology|volume=99|issue=2|pages=656–66|citeseerx=10.1.1.129.504|doi=10.1152/jn.01107.2007|pmid=18057107}} In numerical simulation the integration is stopped if the membrane potential hits an arbitrary threshold (much larger than $V\_T$) at which the membrane potential is reset to a value {{math|V_r}} . The voltage reset value {{math|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. 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.{{Cite journal|last=Richardson|first=Magnus J. E.|date=2007-08-20|title=Firing-rate response of linear and nonlinear integrate-and-fire neurons to modulated current-based and conductance-based synaptic drive|url=https://link.aps.org/doi/10.1103/PhysRevE.76.021919|journal=Physical Review E|volume=76|issue=2|pages=021919|doi=10.1103/PhysRevE.76.021919|pmid=17930077|bibcode=2007PhRvE..76b1919R |url-access=subscription}}

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 [https://neuronaldynamics.epfl.ch/online/Ch5.S2.html Chapter 5.2] of the textbook Neuronal Dynamics.{{Cite book|last=Gerstner, Wulfram|url=https://www.worldcat.org/oclc/885338083|title=Neuronal dynamics : from single neurons to networks and models of cognition|others=Kistler, Werner M., 1969-, Naud, Richard, Paninski, Liam|isbn=978-1-107-44761-5|location=Cambridge|oclc=885338083}}

File:Initial bursting AdEx model.png

The adaptive exponential integrate-and-fire neuron {{cite journal|vauthors=Brette R, Gerstner W|date=November 2005|title=Adaptive exponential integrate-and-fire model as an effective description of neuronal activity|url=https://journals.physiology.org/doi/full/10.1152/jn.00686.2005|journal=Journal of Neurophysiology|volume=94|issue=5|pages=3637–42|doi=10.1152/jn.00686.2005|pmid=16014787}} (AdEx) is a two-dimensional spiking neuron model where the above exponential nonlinearity of the voltage equation is combined with an adaptation variable w

$\backslash tau\_m\; \backslash frac\{dV\}\{dt\}\; =\; R\; I(t)\; +\; [E\_m-V+\backslash Delta\_T\; \backslash exp\; \backslash left(\; \backslash frac\{V\; -\; V\_T\}\; \{\backslash Delta\_T\}\; \backslash right)]\; -\; R\; w$

$\backslash tau\; \backslash frac\{d\; w\; (t)\}\{d\; t\}\; =\; -\; a\; [V\_\backslash mathrm\{m\}\; (t)\; -\; E\_\backslash mathrm\{m\}\; ]-\; w\; +\; b\; \backslash tau\; \backslash delta\; (t-t^f)$

where {{math|w}} denotes an adaptation current with time scale $\backslash tau$. Important model parameters are the voltage reset value {{math|V_r}}, the intrinsic threshold $V\_T$, the time constants $\backslash tau$ and $\backslash tau\_m$ as well as the coupling parameters {{math|a}} and {{math|b}}. The adaptive exponential integrate-and-fire model inherits the experimentally derived voltage nonlinearity 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.{{cite journal|vauthors=Naud R, Marcille N, Clopath C, Gerstner W|date=November 2008|title=Firing patterns in the adaptive exponential integrate-and-fire model|url= |journal=Biological Cybernetics|volume=99|issue=4–5|pages=335–47|doi=10.1007/s00422-008-0264-7|pmc=2798047|pmid=19011922}}

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; and (iii) the broad spectrum of single-neuron firing patterns that can be described by an appropriate choice of AdEx model parameters. 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.

A didactic review of the adaptive exponential integrate-and-fire model (including examples of single-neuron firing patterns) can be found in [https://neuronaldynamics.epfl.ch/online/Ch6.S1.html Chapter 6.1] of the textbook Neuronal Dynamics.

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

$\backslash tau\_m\; \backslash frac\{dV\}\{dt\}\; =\; R\; I(t)\; +\; [E\_m-V+\backslash Delta\_T\; \backslash exp\; \backslash left(\; \backslash frac\{V\; -\; V\_T\}\; \{\backslash Delta\_T\}\; \backslash right)]\; -\; b\; \backslash ,\; [E\_x-V]\; x$

$\backslash tau\_x(V)\; \backslash frac\{d\; x\; (t)\}\{d\; t\}\; =\; x\_0(V\_\backslash mathrm\{m\}\; (t))-\; x$

where b is a coupling parameter, $\backslash tau\_x(V)$ is a voltage-dependent time constant, and $x\_0(V)$ is a saturating nonlinearity, similar to the gating variable m of the Hodgkin-Huxley model. The term $b\; [E\_x-V]\; x$ in the first equation can be considered as a slow voltage-activated ion current.

The GEM is remarkable for two aspects: (i) the nonlinearity of the voltage equation is extracted from experiments; 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.

A review of the computational properties of the GEM and its relation to other spiking neuron models can be found in.{{Cite journal|last1=Brunel|first1=Nicolas|last2=Hakim|first2=Vincent|last3=Richardson|first3=Magnus JE|date=2014-04-01|title=Single neuron dynamics and computation|url=http://www.sciencedirect.com/science/article/pii/S0959438814000130|journal=Current Opinion in Neurobiology|series=Theoretical and computational neuroscience|language=en|volume=25|pages=149–155|doi=10.1016/j.conb.2014.01.005|pmid=24492069|s2cid=16362651|issn=0959-4388|url-access=subscription}}

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Category:Computational neuroscience

Category:Ion channels

Category:Electrophysiology

Category:Nonlinear systems