Exponential integrate-and-fire

Last updated February 28, 2024

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:

{\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)]

.

where $V$ is the membrane potential, $V_{T}$ is the intrinsic membrane potential threshold, $\tau _{m}$ is the membrane time constant, $E_{m}$ is the resting potential, and $\Delta _{T}$ is the sharpness of action potential initiation, usually around 1 mV for cortical pyramidal neurons.^[4] Once the membrane potential crosses $V_{T}$ , 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 $V_{T}$ ) 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)

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

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

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

where $w$ denotes an adaptation current with time scale $\tau$ . Important model parameters are the voltage reset value $V r$ , the intrinsic threshold $V_{T}$ , the time constants $\tau$ and $\tau _{m}$ 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

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

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

where b is a coupling parameter, $\tau _{x}(V)$ 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.^[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]

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References

1 2 3 Fourcaud-Trocmé, Nicolas; Hansel, David; van Vreeswijk, Carl; Brunel, Nicolas (2003-12-17). "How Spike Generation Mechanisms Determine the Neuronal Response to Fluctuating Inputs". The Journal of Neuroscience. 23 (37): 11628–11640. doi:10.1523/JNEUROSCI.23-37-11628.2003. ISSN 0270-6474. PMC 6740955 . PMID 14684865.
1 2 Brette R, Gerstner W (November 2005). "Adaptive exponential integrate-and-fire model as an effective description of neuronal activity". Journal of Neurophysiology. 94 (5): 3637–42. doi:10.1152/jn.00686.2005. PMID 16014787.
1 2 3 4 Richardson, Magnus J. E. (2009-08-24). "Dynamics of populations and networks of neurons with voltage-activated and calcium-activated currents". Physical Review E. 80 (2): 021928. Bibcode:2009PhRvE..80b1928R. doi:10.1103/PhysRevE.80.021928. ISSN 1539-3755. PMID 19792172.
1 2 3 4 5 6 7 8 Badel L, Lefort S, Brette R, Petersen CC, Gerstner W, Richardson MJ (February 2008). "Dynamic I-V curves are reliable predictors of naturalistic pyramidal-neuron voltage traces". Journal of Neurophysiology. 99 (2): 656–66. CiteSeerX 10.1.1.129.504 . doi:10.1152/jn.01107.2007. PMID 18057107.
↑ Ostojic S, Brunel N, Hakim V (August 2009). "How connectivity, background activity, and synaptic properties shape the cross-correlation between spike trains". The Journal of Neuroscience. 29 (33): 10234–53. doi:10.1523/JNEUROSCI.1275-09.2009. PMC 6665800 . PMID 19692598.
↑ Richardson, Magnus J. E. (2007-08-20). "Firing-rate response of linear and nonlinear integrate-and-fire neurons to modulated current-based and conductance-based synaptic drive". Physical Review E. 76 (2): 021919. Bibcode:2007PhRvE..76b1919R. doi:10.1103/PhysRevE.76.021919. PMID 17930077.
1 2 Gerstner, Wulfram. Neuronal dynamics : from single neurons to networks and models of cognition. Kistler, Werner M., 1969-, Naud, Richard, Paninski, Liam. Cambridge. ISBN 978-1-107-44761-5. OCLC 885338083.
1 2 3 Naud R, Marcille N, Clopath C, Gerstner W (November 2008). "Firing patterns in the adaptive exponential integrate-and-fire model". Biological Cybernetics. 99 (4–5): 335–47. doi:10.1007/s00422-008-0264-7. PMC 2798047 . PMID 19011922.
↑ Brunel, Nicolas; Hakim, Vincent; Richardson, Magnus JE (2014-04-01). "Single neuron dynamics and computation". Current Opinion in Neurobiology. Theoretical and computational neuroscience. 25: 149–155. doi:10.1016/j.conb.2014.01.005. ISSN 0959-4388. PMID 24492069. S2CID 16362651.

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[:0-1] 1 2 3 Fourcaud-Trocmé, Nicolas; Hansel, David; van Vreeswijk, Carl; Brunel, Nicolas (2003-12-17). "How Spike Generation Mechanisms Determine the Neuronal Response to Fluctuating Inputs". The Journal of Neuroscience. 23 (37): 11628–11640. doi:10.1523/JNEUROSCI.23-37-11628.2003. ISSN 0270-6474. PMC 6740955 . PMID 14684865.

[:3-2] 1 2 Brette R, Gerstner W (November 2005). "Adaptive exponential integrate-and-fire model as an effective description of neuronal activity". Journal of Neurophysiology. 94 (5): 3637–42. doi:10.1152/jn.00686.2005. PMID 16014787.

[:4-3] 1 2 3 4 Richardson, Magnus J. E. (2009-08-24). "Dynamics of populations and networks of neurons with voltage-activated and calcium-activated currents". Physical Review E. 80 (2): 021928. Bibcode:2009PhRvE..80b1928R. doi:10.1103/PhysRevE.80.021928. ISSN 1539-3755. PMID 19792172.

[badel-4] 1 2 3 4 5 6 7 8 Badel L, Lefort S, Brette R, Petersen CC, Gerstner W, Richardson MJ (February 2008). "Dynamic I-V curves are reliable predictors of naturalistic pyramidal-neuron voltage traces". Journal of Neurophysiology. 99 (2): 656–66. CiteSeerX 10.1.1.129.504 . doi:10.1152/jn.01107.2007. PMID 18057107.

[Ostojic09-5] Ostojic S, Brunel N, Hakim V (August 2009). "How connectivity, background activity, and synaptic properties shape the cross-correlation between spike trains". The Journal of Neuroscience. 29 (33): 10234–53. doi:10.1523/JNEUROSCI.1275-09.2009. PMC 6665800 . PMID 19692598.

[6] Richardson, Magnus J. E. (2007-08-20). "Firing-rate response of linear and nonlinear integrate-and-fire neurons to modulated current-based and conductance-based synaptic drive". Physical Review E. 76 (2): 021919. Bibcode:2007PhRvE..76b1919R. doi:10.1103/PhysRevE.76.021919. PMID 17930077.

[:5-7] 1 2 Gerstner, Wulfram. Neuronal dynamics : from single neurons to networks and models of cognition. Kistler, Werner M., 1969-, Naud, Richard, Paninski, Liam. Cambridge. ISBN 978-1-107-44761-5. OCLC 885338083.

[:2-8] 1 2 3 Naud R, Marcille N, Clopath C, Gerstner W (November 2008). "Firing patterns in the adaptive exponential integrate-and-fire model". Biological Cybernetics. 99 (4–5): 335–47. doi:10.1007/s00422-008-0264-7. PMC 2798047 . PMID 19011922.

[9] Brunel, Nicolas; Hakim, Vincent; Richardson, Magnus JE (2014-04-01). "Single neuron dynamics and computation". Current Opinion in Neurobiology. Theoretical and computational neuroscience. 25: 149–155. doi:10.1016/j.conb.2014.01.005. ISSN 0959-4388. PMID 24492069. S2CID 16362651.

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Exponential integrate-and-fire

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