ExpIFLTC

class brainpy.dyn.ExpIFLTC(size, sharding=None, keep_size=False, mode=None, name=None, spk_fun=InvSquareGrad(alpha=100.0), spk_dtype=None, spk_reset='soft', detach_spk=False, method='exp_auto', init_var=True, scaling=None, V_rest=-65.0, V_reset=-68.0, V_th=-55.0, V_T=-59.9, delta_T=3.48, R=1.0, tau=10.0, V_initializer=ZeroInit, noise=None)

Exponential integrate-and-fire neuron model with liquid time-constant.

Model Descriptions

In the exponential integrate-and-fire model [1], the differential equation for the membrane potential is given by

\[\begin{split}\tau\frac{d V}{d t}= - (V-V_{rest}) + \Delta_T e^{\frac{V-V_T}{\Delta_T}} + RI(t), \\ \text{after} \, V(t) \gt V_{th}, V(t) = V_{reset} \, \text{last} \, \tau_{ref} \, \text{ms}\end{split}\]

This equation has an exponential nonlinearity with “sharpness” parameter \(\Delta_{T}\) and “threshold” \(\vartheta_{rh}\).

The moment when the membrane potential reaches the numerical threshold \(V_{th}\) defines the firing time \(t^{(f)}\). After firing, the membrane potential is reset to \(V_{rest}\) and integration restarts at time \(t^{(f)}+\tau_{\rm ref}\), where \(\tau_{\rm ref}\) is an absolute refractory time. If the numerical threshold is chosen sufficiently high, \(V_{th}\gg v+\Delta_T\), its exact value does not play any role. The reason is that the upswing of the action potential for \(v\gg v +\Delta_{T}\) is so rapid, that it goes to infinity in an incredibly short time. The threshold \(V_{th}\) is introduced mainly for numerical convenience. For a formal mathematical analysis of the model, the threshold can be pushed to infinity.

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. [3]. It is one of the prominent examples of a precise theoretical prediction in computational neuroscience that was later confirmed by experimental neuroscience.

Two important remarks:

• (i) The right-hand side of the above equation contains a nonlinearity that can be directly extracted from experimental data [3]. 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 [4].

References

[1] (1,2)

Fourcaud-Trocmé, Nicolas, et al. “How spike generation mechanisms determine the neuronal response to fluctuating inputs.” Journal of Neuroscience 23.37 (2003): 11628-11640.

[2]

Gerstner, W., Kistler, W. M., Naud, R., & Paninski, L. (2014). Neuronal dynamics: From single neurons to networks and models of cognition. Cambridge University Press.

[3] (1,2)

Badel, Laurent, Sandrine Lefort, Romain Brette, Carl CH Petersen, Wulfram Gerstner, and Magnus JE Richardson. “Dynamic IV curves are reliable predictors of naturalistic pyramidal-neuron voltage traces.” Journal of Neurophysiology 99, no. 2 (2008): 656-666.

[4]

Richardson, Magnus JE. “Firing-rate response of linear and nonlinear integrate-and-fire neurons to modulated current-based and conductance-based synaptic drive.” Physical Review E 76, no. 2 (2007): 021919.

[5]

https://en.wikipedia.org/wiki/Exponential_integrate-and-fire

Examples

There is a simple usage example:

import brainpy as bp

neu = bp.dyn.ExpIFLTC(1, )

# example for section input
inputs = bp.inputs.section_input([0., 5., 0.], [100., 300., 100.])

runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)

bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], show=True)

Model Parameters

Parameter	Init Value	Unit	Explanation
V_rest	-65	mV	Resting potential.
V_reset	-68	mV	Reset potential after spike.
V_th	-30	mV	Threshold potential of spike.
V_T	-59.9	mV	Threshold potential of generating action potential.
delta_T	3.48		Spike slope factor.
R	1		Membrane resistance.
tau	10		Membrane time constant. Compute by R * C.
tau_ref	1.7		Refractory period length.

Model Variables

Variables name	Initial Value	Explanation
V	0	Membrane potential.
input	0	External and synaptic input current.
spike	False	Flag to mark whether the neuron is spiking.
refractory	False	Flag to mark whether the neuron is in refractory period.
t_last_spike	-1e7	Last spike time stamp.

update(x=None)

The function to specify the updating rule.