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)
[edit]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:
- .
![](http://upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Exponential_Integrate-and-Fire_%28schematic%29.jpg/220px-Exponential_Integrate-and-Fire_%28schematic%29.jpg)
where is the membrane potential, is the intrinsic membrane potential threshold, is the membrane time constant, is the resting potential, and is the sharpness of action potential initiation, usually around 1 mV for cortical pyramidal neurons.[4] Once the membrane potential crosses , 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 ) at which the membrane potential is reset to a value Vr . The voltage reset value Vr 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)
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Initial_bursting_AdEx_model.png/220px-Initial_bursting_AdEx_model.png)
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
where w denotes an adaptation current with time scale . Important model parameters are the voltage reset value Vr, the intrinsic threshold , the time constants and 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)
[edit]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
where b is a coupling parameter, is a voltage-dependent time constant, and is a saturating nonlinearity, similar to the gating variable m of the Hodgkin-Huxley model. The term 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]
References
[edit]- ^ a b c 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.
- ^ a b 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.
- ^ a b c d 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.
- ^ a b c d e f g h 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.
- ^ a b 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.
- ^ a b c 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.