Spike frequency adaptation
Boris Gutkin and Fleur Zeldenrust (2014), Scholarpedia, 9(2):30643. | doi:10.4249/scholarpedia.30643 | revision #143322 [link to/cite this article] |
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. Figure 1 B). This phenomenon is called spike-frequency adaptation. 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.
Contents |
Biophysical mechanisms
There are several biophysical mechanisms that can cause spike-frequency adaptation (for a summary, see 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 \(I_M\) and the slow delayed rectifier potassium current \(I_{Ks}\) are slowly activated upon depolarization. These currents can be activated below the spike threshold. \(I_M\) 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. Figure 1 B) current \(I_\textrm{AHP}\) 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 (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 (Spain et al. 1992).
Example models
All three mechanisms discussed above were summarized into a single phenomenological framework by Benda and Herz, (2003), where spike-frequency adaptation is modelled as a shift and distortion (\(\gamma\)) of the \(f-I\) curve caused by a single current \(I_\textrm{adap}\) with the following properties\[\begin{split} I_\textrm{adap} (f) &= A \left(1+\gamma( f )\right),\\ \tau_\textrm{adap} \left(1+\epsilon ( f ) \right) \frac{dA}{dt} & = A_\infty (f) - A, \\ f &= f_0 \left(I- I_\textrm{adap} (f)\right), \end{split} \] where \(A\) is the `adaptation state', that decays with a time constant \(\tau_\textrm{adap} \) to a steady state \(A_\infty (f) \) that depends on the firing frequency of the cell \(f\). The functions \(\gamma (f)\) and \(\epsilon (f)\) 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 \(A\) and in \(\gamma (f)\) 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:
- Linear adaptation in simple models
- A linear mechanism of adaptation was first introduced by Alessandro Treves in the context of a Leaky Integrate-and-Fire neuron\[\begin{split} C \frac{d V}{dt} &= g (V-V_\textrm{rest})+g_k (V-V_k)+I_{stim},\\ \frac{d g_k}{dt} & = -\frac{g_k}{\tau_k} + \Delta g_k \sum_\textrm{spikes} \delta (t-t_\textrm{spike} ). \end{split} \] This model was further simplified towards a rate model by amongst others Treves (1993), Hansel and Sompolinsky (1998) and Fuhrmann et al. (2002) and analyzed using a mean-field approach.
- The Quadratic Integrate-and-Fire (QIF) neuron, or \(\theta\) or Ermentrout-Kopell neuron in its canonical form (Ermentrout, 1996, Ermentrout and Kopell, 1996, Hoppensteadt and Izhikevich, 1997 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 Izhikevich model (Izhikevich, 2003, 2004, 2005, ModelDB)\[\begin{split} C \frac{d V}{dt} &= k (V-V_\textrm{rest})(V-V_\textrm{threshold}) -u+I_{stim},\\ \frac{du}{dt} & = a \left( b \left( V - V_\textrm{rest} \right) -u \right), \\ \textrm{if } V \geq V_\textrm{peak}: & \begin{cases} V &\leftarrow c ,\\ u &\leftarrow u + d. \end{cases} \end{split} \] In this model, spike-dependent adaptation is modelled by parameter \(d\), whereas voltage-dependent adaptation is modelled by parameter \(b\). Brette and Gerstner (Gerstner and Brette, 2009, Brette and Gerstner, 2005) added an equivalent linear model for spike- and voltage-dependent adaptation to the exponential integrate-and-fire model (Fourcaud-Trocmé 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 here.
- Nonlinear adaptation in simple models A more general model for adaptation was added to the QIF neuron by Gutkin et al. (2005)\[\begin{split} \tau_s \frac{d \theta}{dt} &= (1-\cos \theta) + (1+ \cos \theta) \left( \alpha + \beta I_{stim}-g_z z \right),\\ \tau \frac{dz}{dt} & = D(\theta) (1-z) -z,\\ D(\theta) &= \kappa \exp \left( -C \left( 1- \cos \left( \theta - \theta_T \right) \right) \right). \end{split} \] Here, the adaptation current is assumed to depend sigmoidally on the membrane potential, which is in better agreement with channel kinetics.
- A-current in the Connor-Stevens model Connor and Stevens (1971) (see also Connor et al. 1977) define a reduced model that includes a potassium current called the A-current in addition to the spike generating sodium, potassium and leak currents\[ \begin{split} C \frac{dV}{dt} & = I_{stim} - I_L - I_{Na} - I_K - I_A, \\ I_A & = g_A (V-V_A), \\ g_A &= \bar{g}_A \alpha^4 \beta, \\ \frac{dx}{dt} &= \frac{ x_\infty - x}{\tau_x}, \\ x & \in \{\alpha, \beta\}. \end{split} \] This A-current activates slowly (\(\tau_\alpha =\) 12 ms) with depolarization, and inactivates even more slowly (\(\tau_\beta =\) 235 ms), thereby causing voltage-dependent adaptation. The steady-state functions \(\{ \alpha, \beta \}_\infty \) were fitted to experimental values. Note that in this model, \(V_A\) is not necessarily equal to the potassium reversal potential.
- Morris-Lecar model with adaptation currents Prescott et al. (2006, 2008, 2008) split the slow recovery current in the Morris-Lecar model ( 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 \(\beta_z\): 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: version 1, version 2
- Reduced multicompartment models In multicompartment models such as the Miles-Traub model (Traub and Miles, 1991, 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 \(\textrm{K}_\textrm{C}\) 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.
- 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)\[\begin{split} I_{AHP} &= g_{AHP} \frac{\textrm{[Ca]} }{30 + \textrm{[Ca]}} (V - E_K),\\ \frac{d \textrm{[Ca]} }{dt} &= -0.002 I_{Ca} -0.0125 \textrm{[Ca]},\\ I_{Ca} &= g_{Ca} m_{\infty}(V) (V-E_{Ca}), \\ m_\infty &= \frac{1}{1 +\exp(-\frac{(V+25)}{5}) }. \end{split} \]
- The Pinsky and Rinzel model (ModelDB) is a two-compartment reduction of the Miles-Traub model. The slow spike-dependent adaptation caused by \(I_\textrm{AHP}\) 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 \(I_{Ca}\) is a high-threshold calcium current that is only active during a spike, like in the Ermentrout model before.
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 (Brette and Gerstner, 2005, Gerstner and Brette, 2005, Jolivet et al. 2004, Kobayashi et al. 2009, Rauch et al. 2003, 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 (Stiefel et al. 2008, 2009), whereas spike-dependent adaptation can slow down the firing frequency, but cannot change the bifurcation structure or stop repetitive firing (Prescott et al. 2006, 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 (Ermentrout and Terman, 2010). This means the neuron can change the bifurcation with which it starts spiking from a saddle-node on an invariant cycle bifurcation ( type 1 excitability or integrator) to a 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 (Izhikevich 2003, 2004, 2005). For example, the combination of an adaptive potassium current such as \(I_M\) or \(I_{AHP}\) with a depolarizing current such as the low-threshold calcium current (\(I_T\)) or a hyperpolarization-activated cation-selective current (\(I_h\)) can lead to subthreshold oscillations (Huguenard and McCormick 1992, Destexhe et al. 1993, for an overview, see Hutcheon and Yarom, 2000), that effectively make the neuron a resonator. When the adaptive currents are blocked, such neurons act as integrators (Prescott et al. 2008).
Adaptive currents are also shown to be important in intrinsic bursting systems (Skinner et al. 1994, Guckenheimer et al. 1997, for an overview, see 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 (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
- 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, Gutkin et al. (2005) showed that in strongly adapting neurons the 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.
- Ermentrout (1998) and Wang (1998) have shown that slow adaptation can linearize a highly non-linear input-frequency curve.
- 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.
- Puccini et al. (2006,, 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.
- Denève (2008) showed that a form of spike-frequency adaptation is crucial for optimal Bayesian inference with spikes.
- 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: van Vreeswijk and Hansel (2001) showed that adaptation can result in network bursts. Fuhrmann et al. (2001) showed that spike frequency adaptation determines a preferred frequency of stimulation. Ermentrout et al. (2001) showed that both voltage-dependent and spike-dependent adaptation can have a strong effect on the 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 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 (Booth and Bose,2001, Zeldenrust and Wadman, 2009, 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
- Benda(2003). A Universal Model for Spike-Frequency Adaptation. Neural Computation 15(11): 2523-2564.
- Benda, J; Maler, M and Longtin, A (2010). Linear Versus Nonlinear Signal Transmission in Neuron Models With Adaptation Currents or Dynamic Thresholds. Journal of Neurophysiology 104: 2806-2820.
- Booth(2001). Neural Mechanisms for Generating Rate and Temporal Codes in Model CA3 Pyramidal Cells. Journal of Neurophysiology 85: 2432-2445.
- Brette(2005). Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. Journal of Neurophysiology 94(5): 3637-42.
- Connor, B Y J A; Walter, D and McKown, R (1971). Prediction of repetitive firing behaviour from voltage clamp data on an isolated neurone soma. Journal of Physiology 213: 31-53.
- Connor(1977). Neural repetitive firing: modifications of the Hodgkin-Huxley axon suggested by experimental results from crustacean axons. Biophysical Journal 18(1): 81-102.
- Denève, S (2008). Bayesian spiking neurons I: inference. Neural Computation 20(1): 91-117.
- Destexhe, A ; Babloyantz, A and Sejnowski, T J (1993). Ionic Mechanisms for Intrinsic Slow Oscillations in Thalamic Relay Neurons Biophysical Journal 65: 1538-1552.
- Ermentrout, B (1996). Type I Membranes, Phase Resetting Curves, and Synchrony. Neural Computation 8: 979-1001.
- Ermentrout, B (1998). Linearization of F-I curves by adaptation. Neural Computation 10(7): 1721-9.
- Ermentrout(1986). Parabolic Bursting in an Excitable System Coupled with a Slow Oscillation SIAM Journal on Applied Mathematics 46(2): 233-253.
- Ermentrout, B ; Kopell, N J; Pascal, M and Gutkin, B (2001). The effects of spike frequency adaptation and negative feedback on the synchronization of neural oscillators. Neural Computation 13(6): 1285-310.
- Fourcaud-Trocmé, N ; Hansel, D ; van Vreeswijk, C and Brunel, N (2003). How spike generation mechanisms determine the neuronal response to fluctuating inputs. The Journal of Neuroscience 23(37): 11628-40.
- Fuhrmann, H ; Markram, H and Tsodyks, M (2002). Spike frequency adaptation and neocortical rhythms. Journal of Neurophysiology 88(2): 761-70.
- Guckenheimer, J ; Harris-Warrick, R ; Peck, J and Willms, A (1997). Bifurcation, bursting, and spike frequency adaptation. Journal of Computational Neuroscience 4(3): 257-77.
- Gutkin, B S ; Ermentrout, G B and Reyes, A D (2005). Phase-response curves give the responses of neurons to transient inputs. Journal of Neurophysiology 94(2): 1623-35.
- Hansel, D and Sompolinsky, H (1998). "Chapter 13 - Modeling Feature Selectivity in Local Cortical Circuits". in Koch, C and Segev, I. Methods in Neural Modeling: From Ions to Networks. MIT Press. pp. 499-567. ISBN 0262112310.
- Huguenard(1992). Simulation of the Currents Involved in Rhythmic Oscillations in Thalamic Relay Neurons Journal of Neurophysiology 68(4): 1373-1383.
- Hutcheon(2000). Resonance, oscillation and the intrinsic frequency preferences of neurons TRENDS in Neurosciences 23(5): 216-222.
- Izhikevich, E M (2003). Simple Model of Spiking Neurons IEEE Transactions on Neural Networks 14(6): 1572-1596.
- Izhikevich, E M (2004). Which model to use for cortical spiking neurons? IEEE Transactions on Neural Networks 15(5): 1063-1070.
- Jolivet, R ; Lewis, T J and Gerstner, W (2004). Generalized integrate-and-fire models of neuronal activity approximate spike trains of a detailed model to a high degree of accuracy. Journal of neurophysiology 92(2): 959-76.
- Kilpatrick(2011). Sparse gamma rhythms arising through clustering in adapting neuronal networks. PLoS Computational Biology 7(11): e1002281.
- Kobayashi, R ; Tsubo, Y and Shinomoto, S (2009). Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold. Frontiers in computational neuroscience 3: 9.
- La Camera, G et al. (2006). Multiple time scales of temporal response in pyramidal and fast spiking cortical neurons. Journal of Neurophysiology 96(6): 3448-64.
- Latham, P E ; Richmond, B J ; Nelson, P G and Nirenberg, S (2000). Intrinsic Dynamics in Neuronal Networks . I . Theory Journal of Neurophysiology 83: 808-827.
- Morris(1981). Voltage oscillations in the barnacle giant muscle fiber Biophysical Journal 35(1): 193-213.
- Pinsky(1994). Intrinsic and network rhythmogenesis in a reduced Traub model for CA3 neurons Journal of Computational Neuroscience 1(1-2): 39-60.
- Prescott, S A ; Ratté, S ; De Koninck, Y and Sejnowski, TJ (2006). Nonlinear interaction between shunting and adaptation controls a switch between integration and coincidence detection in pyramidal neurons. The Journal of Neuroscience 36(26): 9084-97.
- Prescott, S A ; De Koninck, Y and Sejnowski, T J (2008). Biophysical Basis for Three Distinct Dynamical Mechanisms of Action Potential Initiation PLoS Computational Biology 10(4): e1000198.
- Prescott, S A ; Ratté, S ; De Koninck, Y and Sejnowski, T J (2008). Pyramidal Neurons Switch From Integrators In Vitro to Resonators Under In Vivo-Like Conditions Journal of Neurophysiology 100: 3030-3042.
- Puccini, G D ; Sanchez-Vives, M V and Compte, A (2006). Selective detection of abrupt input changes by integration of spike-frequency adaptation and synaptic depression in a computational network model. Journal of physiology, Paris 100(1-3): 1-15.
- Puccini, G D ; Sanchez-Vives, M V and Compte, A (2007). Integrated mechanisms of anticipation and rate-of-change computations in cortical circuits. PLoS Computational Biology 5(3): e82.
- Rauch, A ; La Camera, G ; Luscher, H-R ; Senn, W and Fusi, S (2003). Neocortical pyramidal cells respond as integrate-and-fire neurons to in vivo-like input currents. Journal of neurophysiology 90(3): 1598-612.
- Rossant, C et al. (2011). Fitting neuron models to spike trains. Frontiers in neuroscience 5: 9.
- Skinner, F K ; Kopell, N and Marder, E (1994). Mechanisms for oscillation and frequency control in reciprocally inhibitory model neural networks. Journal of computational neuroscience 1 (1-2): 69-87.
- Spain, W J ; Schwindt, P C and Crill, W E (1991). Two transient potassium currents in layer V pyramidal neurones from cat sensorimotor cortex Journal of Physiology 434: 591-607.
- Stiefel, K M ; Gutkin, B S and Sejnowski, T J (2008). Cholinergic neuromodulation changes phase response curve shape and type in cortical pyramidal neurons. PLoS One 12(3): e3947.
- Stiefel, K M ; Gutkin, B S and Sejnowski, T J (2009). The effects of cholinergic neuromodulation on neuronal phase-response curves of modeled cortical neurons. Journal of Computational Neuroscience 26(2): 289--301.
- Traub, R D ; Wong, R K S ; Miles, R and Michelson, H (1991). A model of a CA3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances Journal of Neurophysiology 66(2): 635-650.
- Treves, A (1993). Mean-field analysis of neuronal spike dynamics Network: Computation in Neural Systems 4: 259-284.
- van Vreeswijk(2001). Patterns of Synchrony in Neural Networks with Spike Adaptation Neural Computation 13: 959-992.
- Wang, X J (1998). Calcium coding and adaptive temporal computation in cortical pyramidal neurons. Journal of Neurophysiology 79(3): 1549-66.
- Zeldenrust(2009). Two forms of feedback inhibition determine the dynamical state of a small hippocampal network. Neural Networks 22(8): 1139-1158.
- Zeldenrust(2013). Modulation of spike and burst rate in a minimal neuronal circuit with feed-forward inhibition. Neural Networks 40: 1-17.
Internal References
- Gerstner(2009). Adaptive exponential integrate-and-fire model Scholarpedia 4(6): 8427.
Further reading
- Ermentrout, B and Terman, D (2010). Mathematical Foundations of Neuroscience Springer, . ISBN-10: 038787707X ISBN-13: 978-0387877075
- Hoppensteadt, F and Izhikevich, E M (2005). Weakly Connected Neural Networks Springer-Verlag, New York. ISBN 0387949488
- Izhikevich, E M (2005). Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting MIT Press, Cambridge, MA. ISBN 978-0-262-09043-8
- Traub, R D and Miles, R (1991). Neuronal Networks of the Hippocampus Cambridge University Press, Cambridge; New York. ISBN 0521364817, 9780521364812
External links
See also
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