Quadratic Integrate-and-Fire model

The Quadratic Integrate-and-Fire model is a slight modification of the integrate-and-fire model . This slight modification has some drastic changes on the dynamics, although the equation is still solvable analytically in order to determine the period of a continuously-spiking neuron. Because of its simplicity, the Quadratic Integrate-and-Fire model and its variations are favorites for mathematical treatments of neural networks.

Equations

The Quadratic Integrate-and-Fire model replaces the -v from the IF model with a v². Since the voltage will increase very rapidly once greater than one, the threshold value of the equation (at which the v term is reset) can be set at an arbitrary high value and then scaled to 1, with a reset value of 0. Thus, whever the variable v reaches a value of 1, the neuron has an action potential, even though there is no "spike" to speak of.

As in the IF model, the variable b is a function of the input voltage, the membrance capacitance, and the resting potential. As the voltage increases, so too does the value of b, and when b > 0 the neuron spikes continuously. Compared with the IF model, the value of v is said to "blow-up" in finite time as it acquires physically implausible values in a short timeframe. As such, the equation is scaled in accordance to the selected threshold value.

Using the simultation

Included in this simulation are four preset dynamics for applied voltage. Select a maximum possible voltage, a period, and a voltage dynamic. Each of the choices from the drop-down 'dynamics' box corresponds to one of the following:

Constant voltage will administer a steady voltage at the maximum level.
Linear will ramp up the voltage from 0 to the maximum over the given period before staying at the constant maximum voltage.
Periodic will oscillate sinusoidally between 0 and maximum voltage with the given period.
Random will hold a constant voltage randomly selected between 0 and the maximum voltage for one-tenth the period.

The smaller plot on the top-right graphs the input voltage over time. The bottom-right plot show the steady-state frequency of spiking. In this case, frequency is defined as the frequency of firing if the applied voltage was held constant at the present value. For smaller values of I, the applied voltage is insufficient to produce a train of action potentials, so the frequency during those times is zero.

Sources

Ermentrout, G. Bard and David H. Terman. Mathematical Foundations of Neuroscience. Springer 2010. New York, NY.

Izheikevich, Eugene M. Dynamical System in Neuroscience: The Geometry of Excitablilty and Bursting. MIT 2010. Cambridge, Mass.

Credits

This simulation was designed in EJS 4.3.7 by Colin Thomson under the guidance of Dr. Wolfgang Christian. This simulation is part of a series of computational neuroscience simulations for an independent study at Davidson College entitled PHY/CSC 397: Advanced Software Design for Science. The source code, as an EJS file, is available on the comPADRE digital library.

Fall 2012