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[[Image:fig1_sfa.jpg|thumb|650px|right|fig1|Simulation of a Leaky Integrate-and-Fire neuron with a simple spike-dependent adaptation mechanism. When stimulated with a square pulse (A), the neuron fires with a frequency that reduces over time (B, E), due to a hyperpolarizing current (C). This can be quantified by looking at the inter-spike interval (coloured lines in B) or frequency (D) as a function of the spike number (D and E). Note the typical sag or after-hyperpolarization in the membrane potential (B) after the pulse has stopped.]]

When stimulated with a square pulse or step, many neurons show a reduction in the firing frequency of their spike response following an initial increase (Fig. <figref>fig1_sfa.jpg</figref> B). This phenomenon is called <strong>spike-frequency adaptation</strong>. In this article several cellular mechanisms leading to spike-frequency adaptation and its consequences will be discussed.

Spike-frequency adaptation should not be confused with sensory adaptation, a change in responsiveness of a neural system when stimulated with a constant sensory stimulus. Sensory adaptation is a more general phenomenon and spike-frequency adaptation is one of several possible mechanisms by which it is implemented in neural systems. Moreover, spike-frequency adaptation is also one of several possible mechanisms by which neurons can adapt to other stimulus statistics, such as the standard deviation or the time constant of fluctuations. These other forms of adaptation are outside the scope of this article. In a network, spike-frequency adaptation can also be caused by feedback inhibition. However, this article will focus on cellular mechanisms of spike-frequency adaptation.

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==Biophysical mechanisms==

There are several biophysical mechanisms that can cause spike-frequency adaptation (for a summary, see [[#Benda2003| Benda and Herz 2003]]). They all include a form of slow negative feedback to the excitability of the cell. They can be summarized by

# '''Inactivation of depolarizing currents'''  The sodium channels responsible for action potential generation inactivate in response to depolarization, and only recover slowly. The result is that after a spike, less sodium channels are available for action potential generation. Therefore the delay to the next spike increases and the spike amplitude decreases.
#  '''Activity dependent activation of slow hyperpolarizing or shunting currents'''
##  '''Voltage dependent activation''' Voltage-dependent potassium currents such as the non-inactivating potassium current <math>I_M</math> and the slow delayed rectifier potassium current <math>I_{Ks}</math> are slowly activated upon depolarization. These currents can be activated below the spike threshold. <math>I_M</math> is of special interest, as it is modulated by acetylcholine amongst other the neuromodulators.
##  '''Spike dependent activation''' Potassium currents such as the calcium-activated potassium current or the after-hyperpolarization (because of the hyperpolarization after activity, see Fig. <figref>fig1_sfa.jpg</figref> B) current <math>I_\textrm{AHP}</math> are activated by the increase in calcium concentration in the cell due to the influx of calcium at the peak of an action potential. Therefore, these currents are only activated after spiking activity. These  currents cause an after-hyperpolarizing potential (AHP) after a spike or burst of spikes.

There is an abundance of data about mechanisms that could lead to spike-frequency adaptation, which is beyond the scope of this article. There is no specific time scale for adaptation ([[#LaCamera2006|La Camera et al. 2006]]): neurons can have several adaptation processes, that have different time constants. Indeed, ionic currents with a wide range of time constants have been reported ([[#Spain1991|Spain et al. 1992]]).

==Example models==

All three mechanisms discussed above were summarized into a single phenomenological framework by [[#Benda2003| Benda and Herz, (2003)]], where spike-frequency adaptation is modelled as a shift and distortion (<math>\gamma</math>) of the <math>f-I</math> curve caused by a single current <math>I_\textrm{adap}</math> with the following properties:
<math>\begin{split} I_\textrm{adap} (f) &=  A  \left(1+\gamma( f )\right),\\
\tau_\textrm{adap} \left(1+\epsilon ( f ) \right) \frac{dA}{dt} & = A_\infty (f) - A, \\ 
f &= f_0 \left(I- I_\textrm{adap} (f)\right), \end{split} </math>
where <math>A</math> is the `adaptation state', that decays with a time constant <math>\tau_\textrm{adap} </math> to a steady state <math>A_\infty (f) </math> that depends on the firing frequency of the cell <math>f</math>. The functions <math>\gamma (f)</math> and <math>\epsilon (f)</math> represent the dependence of the adaptation current and the time constant of adaptation on the frequency. This framework is purely phenomenological, in that it describes only the effects on and as a result of the firing frequency of the neuron. Voltage-dependent mechanisms are included in parameter <math>A</math> and in <math>\gamma (f)</math> by averaging the effects of voltage-dependent mechanisms over a single inter-spike interval. 

Spike-frequency adaptation has been modeled in all three forms mentioned above, both in extended and in simplified models, including (but not limited to) the following models:

<ul>
<li>'''Linear adaptation in simple models'''

<ul><li>A linear mechanism of adaptation was first introduced by [[#Treves1993| Alessandro Treves]] in the context of a Leaky Integrate-and-Fire neuron:
<math>\begin{split} C \frac{d V}{dt} &= g (V-V_\textrm{rest})+g_k (V-V_k)+I_{stim},\\
\frac{d g_k}{dt} & = -\frac{g_k}{\tau_k} + \Delta g_k \sum_\textrm{spikes} \delta (t-t_\textrm{spike} ).
\end{split}
</math> 
This model was further simplified towards a rate model by amongst others [[#Treves1993| Treves (1993)]], [[#Hansel1998| Hansel and Sompolinsky (1998)]] and [[#Fuhrmann2002| Fuhrmann et al. (2002)]] and analyzed using a mean-field approach. </li>

<li> The [http://www.scholarpedia.org/article/Quadratic_integrate_and_fire_neuron Quadratic Integrate-and-Fire (QIF)] neuron, or <math>\theta</math> or [http://www.scholarpedia.org/article/Ermentrout-Kopell_canonical_model Ermentrout-Kopell neuron] in its canonical form ([[#Ermentrout1996|Ermentrout, 1996]],[[#Ermentrout1986| Ermentrout and Kopell, 1996]],[[#Hoppensteadt1997| Hoppensteadt and Izhikevich, 1997]][[#Latham2000| Latham et al., 2000]]), has a threshold that decreases with the amplitude of the injected current. It can be augmented with a linear adaptation mechanism, in which case it becomes equivalent to the [http://www.scholarpedia.org/article/Izhikevich_neuron Izhikevich model] ([[#Izhikevich2003a|Izhikevich, 2003]],[[#Izhikevich2004 | 2004]],[[#Izhikevich2005| 2005]], [http://senselab.med.yale.edu/modeldb/ShowModel.asp?model=39948 ModelDB]):
<math>\begin{split} C \frac{d V}{dt} &= k (V-V_\textrm{rest})(V-V_\textrm{threshold}) -u+I_{stim},\\
\frac{du}{dt} & = a \left( b \left( V - V_\textrm{rest} \right) -u \right), \\
\textrm{if } V \geq V_\textrm{peak}: & \begin{cases}  V &\leftarrow c ,\\
u &\leftarrow u + d.
\end{cases}
\end{split}
</math> 
In this model, spike-dependent adaptation is modelled by parameter <math>d</math>, whereas voltage-dependent adaptation is modelled by parameter <math>b</math>. Brette and Gerstner ([[#Gerstner2009|Gerstner and Brette, 2009]],[[#Brette2005| Brette and Gerstner, 2005]]) added an equivalent linear model for spike- and voltage-dependent adaptation to the [http://www.scholarpedia.org/article/Adaptive_exponential_integrate-and-fire_model exponential integrate-and-fire model] ([[#Fourcaud-Trocme2003|Fourcaud-Trocm&eacute; et al. 2003]]), a neuron model in which the membrane potential evolves exponentially. In both models, voltage-dependent adaptation is assumed to depend linearly on the membrane potential. The models with adaptation have an equivalent bifurcation structure; a discussion on the main differences and similarities of the models is presented [http://www.scholarpedia.org/article/Adaptive_exponential_integrate-and-fire_model here]. </li>
</ul>
</li>

<li> '''Nonlinear adaptation in simple models''' A more general model for adaptation was added to the QIF neuron by [[#Gutkin2005|Gutkin et al. (2005)]]: 
<math>\begin{split} \tau_s \frac{d \theta}{dt} &= (1-\cos \theta) + (1+ \cos \theta) \left( \alpha + \beta I_{stim}-g_z z \right),\\
\tau \frac{dz}{dt} & = D(\theta) (1-z) -z,\\
D(\theta) &= \kappa \exp \left( -C \left( 1- \cos \left( \theta - \theta_T \right) \right) \right).
\end{split}
</math> 
Here, the adaptation current is assumed to depend sigmoidally on the membrane potential, which is in better agreement with channel kinetics. </li>

<li>'''A-current in the Connor-Stevens model''' [[#Connor1971| Connor and Stevens (1971)]] (see also [[#Connor1977| Connor et al. 1977]]) define a reduced [http://www.scholarpedia.org/article/Connor-Stevens_model model] that includes a potassium current called the [http://www.scholarpedia.org/article/A-current A-current] in addition to the spike generating sodium, potassium and leak currents:<math> \begin{split}
C \frac{dV}{dt} & = I_{stim} - I_L - I_{Na} - I_K - I_A,  \\
I_A & = g_A (V-V_A), \\
g_A &= \bar{g}_A \alpha^4 \beta, \\
\frac{dx}{dt}  &= \frac{ x_\infty - x}{\tau_x}, \\
x & \in \{\alpha, \beta\}.
\end{split}
</math>
This A-current activates slowly (<math>\tau_\alpha =</math> 12 ms) with depolarization, and inactivates even more slowly (<math>\tau_\beta =</math> 235 ms), thereby causing voltage-dependent adaptation. The steady-state functions <math>\{ \alpha, \beta \}_\infty </math> were fitted to experimental values. Note that in this model, <math>V_A</math> is not necessarily equal to the potassium reversal potential. </li>

<li> '''Morris-Lecar model with adaptation currents''' Prescott et al. ([[#Prescott2006|2006]],[[#Prescott2008| 2008]],[[#Prescott2008a| 2008]]) split the slow recovery current in the [http://www.scholarpedia.org/article/Morris-Lecar_model Morris-Lecar model] ([[#Morris1981| Morris and Lecar, 1981]]) into the delayed-rectifier potassium current and adaptive currents, to investigate the effects of different forms of adaptation on the the type of bifurcation by which the neuron loses the resting state and starts spiking. The model now reads
\[\begin{split}
C \frac{dV}{dt} &= I_{stim} - \bar{g}_{fast} m_\infty (V-E_{Na}) - \bar{g}_{K,Dr} y (V-E_K)-\bar{g}_{adap} z (V-E_{adap}),\\
\frac{dz}{dt} & = \phi_z \frac{z_\infty (V) - z}{\tau_z (V)},\\
z_\infty (V) & = 0.5 [1 + \tanh\left(\frac{V-\beta_z}{\gamma_z} \right) ], \\
\tau_z (V) & = 1 / \cosh \left( \frac{V-\beta_z}{2 \gamma_z}\right).
\end{split}
\]
The adaptation current can have voltage-dependent activation or spike-dependent activation, depending on the choice of the parameter <math>\beta_z</math>: a relatively high value (0 mV) gives spike-dependent activation and a relatively low value (-35 mV) voltage-dependent activation. Different versions can be found in the ModelDB: [http://senselab.med.yale.edu/modeldb/showmodel.asp?model=116123 version 1], [http://senselab.med.yale.edu/modeldb/showmodel.asp?model=116386 version 2] </li>

<li>'''Reduced multicompartment models''' In multicompartment models such as the Miles-Traub model ([[#Traub1991b|Traub and Miles, 1991]],[[#Traub1991| Traub et al., 1991]]), a 19-compartment model of a hippocampal CA-3 pyramidal neuron, adaptation is a result of several adaptive currents. These include the A-current and AHP-current mentioned before and the <math>\textrm{K}_\textrm{C}</math> current, a potassium current that depends on both the membrane potential and the calcium concentration within the cell, and shows inactivation. A final source of adaptation in the Miles-Traub model is given by the inactivation of sodium channels. This model has been reduced to single- and two-compartment models. 
<ul>
<li> [[#Ermentrout1998|Ermentrout (1998)]] reduced the Miles-Traub model to a single-compartment model. They showed that slow adaptation linearizes the frequency-current curve. In this model, the slow spike-dependent adaptation activates instantaneously as a function of the calcium concentration [Ca] (which has slow dynamics):
<math>\begin{split}
I_{AHP} &= g_{AHP} \frac{\textrm{[Ca]} }{30 + \textrm{[Ca]}} (V - E_K),\\
\frac{d \textrm{[Ca]} }{dt} &= -0.002 I_{Ca} -0.0125 \textrm{[Ca]},\\
I_{Ca} &= g_{Ca} m_{\infty}(V) (V-E_{Ca}), \\
m_\infty &= \frac{1}{1 +\exp(-\frac{(V+25)}{5}) }.
\end{split}
</math> </li>
<li>The [[#Pinsky1994|Pinsky and Rinzel model]] ([http://senselab.med.yale.edu/modeldb/showmodel.asp?model=35358 ModelDB]) is a two-compartment reduction of the Miles-Traub model. The slow spike-dependent adaptation caused by <math>I_\textrm{AHP}</math> in the dendrite was shown to be crucial for the slow bursting state. This current is modelled as a potassium current which opens with the calcium concentration
\[\begin{split}
I_{AHP} &= \bar{g}_{AHP} q (V-V_K), \\
\frac{dq}{dt}& = \alpha_q (1-q) -\beta_q q,\\
\alpha_q &= \min((0.00002)\textrm{[Ca]}, 0.01), \\
\beta_q &=0.001,\\
\frac{d\textrm{[Ca]}}{dt} &= -0.13 I_{Ca} - 0.075\textrm{[Ca]},
\end{split}
\]
where <math>I_{Ca}</math> is a high-threshold calcium current that is only active during a spike, like in the Ermentrout model before.</li>
</ul>
</ul>
It has been reported by several authors that spike responses of cortical neurons can be adequately fitted by a very simple neuron model, as long as an adaptation mechanism is included ([[#Brette2005|Brette and Gerstner, 2005]],[[#Gerstner2009| Gerstner and Brette, 2005]],[[#Jolivet2004| Jolivet et al. 2004]],[[#Kobayashi2009| Kobayashi et al. 2009]],[[#Rauch2003| Rauch et al. 2003]],[[#Rossant2011| Rossant et al. 2011]]).

==Consequences==

Spike-frequency adaptation has several consequences for the spiking behaviour of the neuron. Here, the dynamical and functional consequences of adaptation on action potential firing will be discussed. 

===Dynamical=== 

Voltage dependent currents can lead to a change in the bifurcation structure of the neuron ([[#Stiefel2008|Stiefel et al. 2008]],[[#Stiefel2009| 2009]]), whereas spike-dependent adaptation can slow down the firing frequency, but cannot change the bifurcation structure or stop repetitive firing ([[#Prescott2006|Prescott et al. 2006]],[[#Prescott2008| 2008]]). With the amplitude of the input current and the strength of adaptation as bifurcation parameters, a codimension 2 bifurcation can be made: the Bogdanov-Takens bifurcation ([[#Ermentrout2010|Ermentrout and Terman, 2010]]). This means the neuron can change the bifurcation with which it starts spiking  from a [http://www.scholarpedia.org/article/Saddle-Node_Bifurcation_on_Invariant_Circle saddle-node on an invariant cycle bifurcation] ( [http://www.scholarpedia.org/article/Neuronal_Excitability type 1 excitability] or integrator) to a [http://www.scholarpedia.org/article/Hopf_bifurcation Andronov-Hopf bifurcation] (type 2 excitability or resonator). The canonical form of a neuron close to a Bogdanov-Takens bifurcation is equivalent to the Izhikevich model ([[#Izhikevich2003a|Izhikevich 2003]],[[#Izhikevich2004| 2004]],[[#Izhikevich2005| 2005]]). For example, the combination of an adaptive potassium current such as <math>I_M</math> or <math>I_{AHP}</math> with a depolarizing current such as the low-threshold calcium current (<math>I_T</math>) or a hyperpolarization-activated cation-selective current (<math>I_h</math>) can lead to subthreshold oscillations ([[#Huguenard1992|Huguenard and McCormick 1992]],[[#Destexhe1993| Destexhe et al. 1993]], for an overview, see [[Hutcheon2000|Hutcheon and Yarom, 2000]]), that effectively make the neuron a resonator. When the adaptive currents are blocked, such neurons act as integrators ([[#Prescott2008|Prescott et al. 2008]]). 

Adaptive currents are also shown to be important in intrinsic [http://www.scholarpedia.org/article/Bursting bursting systems] ([[#Skinner1994|Skinner et al. 1994]], [[#Guckenheimer1997|Guckenheimer et al. 1997]], for an overview, see [[#Izhikevich2005|Izhikevich, 2005]]): slow adaptation is one of the mechanisms to terminate a the spiking phase, thereby terminating the burst. Moreover, with strong excitatory coupling, adaptive currents can induce network bursts ([[#VanVreeswijk2001|van Vreeswijk and Hansel, 2001]])

===Functional===

Adaptive mechanisms play a crucial role in the encoding of stimulus parameters by the output spike train. These effects of spike-frequency adaptation on coding include

* [http://www.scholarpedia.org/article/Neuronal_Excitability Integrators and resonators] encode information in a fundamentally different way, because they respond to different features in the input. Therefore, the aforementioned switch from type 1 to type 2 by spike-frequency adaptation changes the coding properties of the neuron, making it more sensitive to synchronous activity. 
* In relation to the shift from integrator to resonator, [[#Gutkin2005|Gutkin et al. (2005)]] showed that in strongly adapting neurons the [http://www.scholarpedia.org/article/Phase_response_curve phase response curves] are skewed to the right at low firing frequencies and can have a negative region, which means that spikes are triggered most by inputs received close to the last spike, making the neuron a coincidence detector. For high firing frequencies, this skewness is decreased, making the neuron more of an integrator. 
* [[#Ermentrout1998|Ermentrout (1998)]] and [[#Wang1998|Wang (1998)]] have shown that slow adaptation can linearize a highly non-linear input-frequency curve.
* [[#Benda2010|Benda et al. (2010)]] showed that spike-frequency adaptation caused by a dynamic threshold and adaptation caused by ionic currents have different effects on the onset f-I curve (the frequency of the spike response directly after the onset of a step-stimulus): adaptive currents have a subtractive effect (they shift the curve towards higher input values), effectively adding a high-pass filter component to the transfer function of the neuron. Alternatively,  a dynamic threshold has a divisive effect on the onset f-I curve (it decreases the slope of the curve), with more complex non-linear effects on the transfer function. 
* With strong enough voltage-dependent adaptation, a neuron can shift to a type 3 neuron, which only gives a transient response to the onset of stimulus. The inactivation of sodium currents can also have this effect. However, spike-dependent adaptation can never turn a neuron into a type 3 neuron.
* [[#Puccini2006|Puccini et al. (2006,]],[[#Puccini2007| 2007)]] showed that spike-frequency adaptation in combination with short-term synaptic depression has as an effect that neurons respond mostly to abrupt changes in the input, so to the temporal derivative or rate-of-change.
* [[#Deneve2008|Den&egrave;ve (2008)]] showed that a form of spike-frequency adaptation is crucial for optimal [http://www.scholarpedia.org/article/Bayesian_inference Bayesian inference] with spikes.
* [[#Kilpatrick2011|Kilpatrick and Ermentrout (2011)]] showed that formation of clusters, a subpopulation of neurons that fire together and regularly for some time, depends crucially on spike-frequency adaptation, thereby influencing population coding.
* Several authors considered synchronization in networks with adaptation: [[#Vreeswijk2001|van Vreeswijk and Hansel (2001)]] showed that adaptation can result in network bursts. [[#Fuhrmann2002| Fuhrmann et al. (2001)]] showed that spike frequency adaptation determines a preferred frequency of stimulation. [[#Ermentrout2001|Ermentrout et al. (2001)]] showed that both voltage-dependent and spike-dependent adaptation can have a strong effect on the [http://www.scholarpedia.org/article/Synchronization synchronization] of a network of excitatory type 1 neurons: adaptation can synchronize excitatory coupled cells that would not synchronize otherwise. 
* As discussed before, adaptive currents can play an important role in the switch from regular firing to (intrinsic) bursting, thereby strongly affecting the neural code. Bursts can encode different features, have more informational content and have many other effects on the neural code (for an overview, see [http://www.scholarpedia.org/article/Bursting Bursting] ). Moreover, adaptive currents can strongly shape the phase response curves of bursting systems in response to inhibition, which results in a subtle interplay between adaptive currents and inhibition in these systems ([[#Booth2001|Booth and Bose,2001]],[[#Zeldenrust2009| Zeldenrust and Wadman, 2009]],[[#Zeldenrust2013| 2013]]).

==Disclaimer==
Spike frequency adaptation is a physiological phenomenon (or even a collection of phenomena) that has had a vast amount of publications addressing it (at least 450 papers by searching PubMed). It is impossible to give a full account of this literature, thus this article focused on the definition, the classical mechanisms, examples of models and a selected sub-set of functional consequences. The readers are invited to contact the authors with suggestions and additions.


==References==
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* {{cite book
| editor1-first = C
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| editor2-last = Segev
| title = Methods in Neural Modeling: From Ions to Networks
| publisher = MIT Press
| chapter = Chapter 13 - Modeling Feature Selectivity in Local Cortical Circuits
| first1 = D
| last1 = Hansel
| first2 = H
| last2 = Sompolinsky
| pages = 499-567
| date = 1998
| isbn= 0262112310 
| ref = Hansel1998}}
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*{{Bibitem article 3|Integrated mechanisms of anticipation and rate-of-change computations in cortical circuits.|PLoS Computational Biology|5(3)|2007|e82|Puccini| G D | Sanchez-Vives| M V | Compte| A|label=Puccini2007}}
*{{Bibitem article 5|Neocortical pyramidal cells respond as integrate-and-fire neurons to in vivo-like input currents.|Journal of neurophysiology|90(3)|2003|1598-612|Rauch| A |La Camera| G | Luscher| H-R | Senn| W | Fusi| S|label=Rauch2003}}
*{{Bibitem article etal|Fitting neuron models to spike trains.|Frontiers in neuroscience|5|2011| 9|Rossant| C|Goodman| D F M|Fontaine| B|Platkiewicz| J|Magnusson| A K| Brette| R|label=Rossant2011}}
*{{Bibitem article 3|Mechanisms for oscillation and frequency control in reciprocally inhibitory model neural networks.|Journal of computational neuroscience|1 (1-2)|1994|69-87|Skinner| F K | Kopell| N | Marder| E|label=Skinner1994}}
*{{Bibitem article 3|Two transient potassium currents in layer V pyramidal neurones from cat sensorimotor cortex|Journal of Physiology|434|1991|591-607|Spain| W J | Schwindt| P C | Crill| W E|label=Spain1991}}
*{{Bibitem article 3|Cholinergic neuromodulation changes phase response curve shape and type in cortical pyramidal neurons.|PLoS One|12(3)|2008|e3947|Stiefel| K M | Gutkin| B S | Sejnowski| T J|label=Stiefel2008}}
*{{Bibitem article 3|The effects of cholinergic neuromodulation on neuronal phase-response curves of modeled cortical neurons.|Journal of Computational Neuroscience|26(2)|2009|289--301|Stiefel| K M | Gutkin| B S | Sejnowski| T J|label=Stiefel2009}}
*{{Bibitem article 4|A model of a CA3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances|Journal of Neurophysiology|66(2)|1991|635-650|Traub| R D | Wong| R K S | Miles| R | Michelson| H|label=Traub1991}}
*{{Bibitem article 1|Mean-field analysis of neuronal spike dynamics|Network: Computation in Neural Systems|4|1993|259-284|Treves| A |label=Treves1993}}
*{{Bibitem article 2|Patterns of Synchrony in Neural Networks with Spike Adaptation|Neural Computation|13|2001|959-992|van Vreeswijk| C| Hansel| D|label=VanVreeswijk2001}}
*{{Bibitem article 1|Calcium coding and adaptive temporal computation in cortical pyramidal neurons.|Journal of Neurophysiology|79(3)|1998|1549-66|Wang| X J |label=Wang1998}}
*{{Bibitem article 2|Two forms of feedback inhibition determine the dynamical state of a small hippocampal network.|Neural Networks|22(8)|2009|1139-1158|Zeldenrust|F|Wadman|W J|label=Zeldenrust2009}}
*{{Bibitem article 2|Modulation of spike and burst rate in a minimal neuronal circuit with feed-forward inhibition.|Neural Networks|40|2013|1-17|Zeldenrust|F|Wadman|W J|label=Zeldenrust2013}}

===Internal References===
*{{Bibitem article 2|Adaptive exponential integrate-and-fire model|Scholarpedia|4(6)|2009|8427|Gerstner| W | Brette| R|label=Gerstner2009}}

== Further reading== 
<!-- Provide a list of recent books that you recommend to read to learn more about your topic. This section may contain less formatted bibliography. It is a good place to cite introductory books, tutorials, and reviews. -->
*{{Bibitem book 2|Mathematical Foundations of Neuroscience|Springer||2010|Ermentrout| B | Terman| D |isbn = ISBN-10: 038787707X ISBN-13: 978-0387877075|label=Ermentrout2010}}
*{{Bibitem book 2|Weakly Connected Neural Networks|Springer-Verlag|New York|2005|Hoppensteadt| F |Izhikevich|E M|isbn = ISBN 0387949488|label=Hoppensteadt1997}}
*{{Bibitem book 1|Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting|MIT Press|Cambridge, MA|2005|Izhikevich|E M|isbn = ISBN 978-0-262-09043-8|label=Izhikevich2005}}
*{{Bibitem book 2|Neuronal Networks of the Hippocampus|Cambridge University Press|Cambridge; New York|1991|Traub|R D|Miles|R|isbn = ISBN 0521364817, 9780521364812|label=Traub1991b}}

==External links==
<!-- make links online tutorials, author websites, etc. -->   

[http://senselab.med.yale.edu/modeldb ModelDB]

==See also==
<!--Provide an alphabetical list of links to other relevant articles in Scholarpedia -->

[[A-current]],
[[Adaptive exponential integrate-and-fire model]],
[[Bayesian inference]], 
[[Bursting]],
[[Connor-Stevens model]], 
[[Ermentrout-Kopell canonical model]],
[[Hopf bifurcation]],
[[Izhikevich neuron]], 
[[Morris-Lecar model]],
[[Neuronal Excitability]],
[[Phase response curve]],
[[Quadratic integrate and fire neuron]],  
[[Saddle-Node Bifurcation on Invariant Circle]]

[[Category:Computational neuroscience]]