[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/2015\/10\/23\/exponential-integrate-and-fire-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/2015\/10\/23\/exponential-integrate-and-fire-wikipedia\/","headline":"Exponential integrate-and-fire &#8211; Wikipedia","name":"Exponential integrate-and-fire &#8211; Wikipedia","description":"Spiking neuron model Exponential integrate-and-fire models are compact and computationally efficient nonlinear spiking neuron models with one or two variables.","datePublished":"2015-10-23","dateModified":"2015-10-23","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/86f691681dd09ecbcdb84ddae90a2e507f19afc1","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/86f691681dd09ecbcdb84ddae90a2e507f19afc1","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/2015\/10\/23\/exponential-integrate-and-fire-wikipedia\/","wordCount":6825,"articleBody":"Spiking neuron modelExponential 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.Table of ContentsExponential\u00a0integrate-and-fire  (EIF) [edit]Adaptive Exponential\u00a0integrate-and-fire  (AdEx) [edit]Generalized Exponential\u00a0integrate-and-fire Model  (GEM) [edit]References[edit]Exponential\u00a0integrate-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-Trocm\u00e9, 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:dVdt\u2212R\u03c4mI(t)=1\u03c4m[Em\u2212V+\u0394Texp\u2061(V\u2212VT\u0394T)]{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 th exponential integrate-and-fire neuron can be extracted from experimental data.[5]where V{displaystyle V} is the membrane potential, VT{displaystyle V_{T}} is the intrinsic membrane potential threshold,  \u03c4m{displaystyle tau _{m}} is the membrane time constant, Em{displaystyle E_{m}}is the resting potential, and \u0394T{displaystyle Delta _{T}} is the sharpness of action potential initiation, usually around 1 mV for cortical pyramidal neurons.[4] Once the membrane potential crosses VT{displaystyle V_{T}}, it diverges to infinity in finite time.[6][5] In numerical simulation the integration is stopped if the membrane potential hits an arbitrary threshold (much larger than VT{displaystyle V_{T}}) 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.[5] 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.[7]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.[8]Adaptive Exponential\u00a0integrate-and-fire  (AdEx) [edit]  Initial bursting AdEx modelThe 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\u03c4mdVdt=RI(t)+[Em\u2212V+\u0394Texp\u2061(V\u2212VT\u0394T)]\u2212Rw{displaystyle tau _{m}{frac {dV}{dt}}=RI(t)+[E_{m}-V+Delta _{T}exp left({frac {V-V_{T}}{Delta _{T}}}right)]-Rw}\u03c4dw(t)dt=\u2212a[Vm(t)\u2212Em]\u2212w+b\u03c4\u03b4(t\u2212tf){displaystyle tau {frac {dw(t)}{dt}}=-a[V_{mathrm {m} }(t)-E_{mathrm {m} }]-w+btau delta (t-t^{f})}where w denotes an adaptation current with time scale \u03c4{displaystyle tau }. Important model parameters are the voltage reset value  Vr, the intrinsic threshold VT{displaystyle V_{T}}, the time constants \u03c4{displaystyle tau } and \u03c4m{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 [5] 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.[9]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;[5] and (iii) the broad spectrum of single-neuron firing patterns that can be described by an appropriate choice of AdEx model parameters.[9]  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.[9]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.[8]Generalized Exponential\u00a0integrate-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\u03c4mdVdt=RI(t)+[Em\u2212V+\u0394Texp\u2061(V\u2212VT\u0394T)]\u2212b[Ex\u2212V]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}\u03c4x(V)dx(t)dt=x0(Vm(t))\u2212x{displaystyle tau _{x}(V){frac {dx(t)}{dt}}=x_{0}(V_{mathrm {m} }(t))-x}where b is a coupling parameter, \u03c4x(V){displaystyle tau _{x}(V)} is a voltage-dependent time constant, and x0(V){displaystyle x_{0}(V)} is a saturating nonlinearity, similar to the gating variable m of the Hodgkin-Huxley model. The term b[Ex\u2212V]x{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;[5] 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.[10]References[edit]^ a b c Fourcaud-Trocm\u00e9, Nicolas; Hansel, David; van Vreeswijk, Carl; Brunel, Nicolas (2003-12-17). &#8220;How Spike Generation Mechanisms Determine the Neuronal Response to Fluctuating Inputs&#8221;. The Journal of Neuroscience. 23 (37): 11628\u201311640. doi:10.1523\/JNEUROSCI.23-37-11628.2003. ISSN\u00a00270-6474. PMC\u00a06740955. PMID\u00a014684865.^ a b Brette R, Gerstner W (November 2005). &#8220;Adaptive exponential integrate-and-fire model as an effective description of neuronal activity&#8221;. Journal of Neurophysiology. 94 (5): 3637\u201342. doi:10.1152\/jn.00686.2005. PMID\u00a016014787.^ a b c d Richardson, Magnus J. E. (2009-08-24). &#8220;Dynamics of populations and networks of neurons with voltage-activated and calcium-activated currents&#8221;. Physical Review E. 80 (2): 021928. doi:10.1103\/PhysRevE.80.021928. ISSN\u00a01539-3755. PMID\u00a019792172.^ a b Badel L, Lefort S, Brette R, Petersen CC, Gerstner W, Richardson MJ (February 2008). &#8220;Dynamic I-V curves are reliable predictors of naturalistic pyramidal-neuron voltage traces&#8221;. Journal of Neurophysiology. 99 (2): 656\u201366. doi:10.1152\/jn.01107.2007. PMID\u00a018057107.^ a b c d e f Badel L, Lefort S, Brette R, Petersen CC, Gerstner W, Richardson MJ (February 2008). &#8220;Dynamic I-V curves are reliable predictors of naturalistic pyramidal-neuron voltage traces&#8221;. Journal of Neurophysiology. 99 (2): 656\u201366. CiteSeerX\u00a010.1.1.129.504. doi:10.1152\/jn.01107.2007. PMID\u00a018057107.^ Ostojic S, Brunel N, Hakim V (August 2009). &#8220;How connectivity, background activity, and synaptic properties shape the cross-correlation between spike trains&#8221;. The Journal of Neuroscience. 29 (33): 10234\u201353. doi:10.1523\/JNEUROSCI.1275-09.2009. PMC\u00a06665800. PMID\u00a019692598.^ Richardson, Magnus J. E. (2007-08-20). &#8220;Firing-rate response of linear and nonlinear integrate-and-fire neurons to modulated current-based and conductance-based synaptic drive&#8221;. Physical Review E. 76 (2): 021919. doi:10.1103\/PhysRevE.76.021919. PMID\u00a017930077.^ a b Gerstner, Wulfram. Neuronal dynamics\u00a0: from single neurons to networks and models of cognition. Kistler, Werner M., 1969-, Naud, Richard, Paninski, Liam. Cambridge. ISBN\u00a0978-1-107-44761-5. OCLC\u00a0885338083.^ a b c Naud R, Marcille N, Clopath C, Gerstner W (November 2008). &#8220;Firing patterns in the adaptive exponential integrate-and-fire model&#8221;. Biological Cybernetics. 99 (4\u20135): 335\u201347. doi:10.1007\/s00422-008-0264-7. PMC\u00a02798047. PMID\u00a019011922.^ Brunel, Nicolas; Hakim, Vincent; Richardson, Magnus JE (2014-04-01). &#8220;Single neuron dynamics and computation&#8221;. Current Opinion in Neurobiology. Theoretical and computational neuroscience. 25: 149\u2013155. doi:10.1016\/j.conb.2014.01.005. ISSN\u00a00959-4388. PMID\u00a024492069. 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