Alla Borisyuk
Math 6770, Computational Neuroscience
Homework 3
1. Consider an LIF model:
C
dV
= −gL (V − VL ) + I,
dt
with reset condition including absolute refractory period V (t−
0 ) = Vth , then V (t) = Vreset for
t0 < t < t0 + tref .
a) Find a sample set of realistic parameter values for this model by looking up appropriate
numbers for your favorite cell type.
b) Find the minimal level of constant current for which the cell will fire repetitively (in this one
I don’t want a number, but rather Icrit as a function of other parameters
c) Compute the f-I curve for this model. It is nice to compute a formula. Numerical result
would be acceptable too. What is the role of the non-zero absolute refractory period in shaping
the firing rate curve?
d) With tref = 0, add the spike-triggered (i.e. the gating variable jumps at each spike) shunting
(i.e. the reversal is equal to VL ) adaptaton current to the model. Find the instantateous f-I relation
(i.e. the inverse of the interspike interval as a function of I) for the first interspike interval, the
second one and the adapted one. How would you describe the effect of the adaptation on the f-I
curve?
2. (I book) The simplest implementation of a pulse-coupled all-to-all non-dimensional integrateand-fire neural network has the form
v̇i = bi − vi +
X
6= icij δ(t − tj ),
j
where tj is the moment of firing of the j-th neuron, that is, the moment vj (tj ) = 1. (In this
non-dimensional form the voltage is relative to rest, i.e. the resting state is at 0, and the threshold
has been normalized to 1) . Thus, whenever the j-th neuron fires, the membrane potentials of the
other neurons are instantaneously adjusted by cij .
a) Show that the same initial conditions may result in different solutions, depending on the
implementation details. In other words, pick initial conditions and parameters (reset, bi and cij ),
see what the network is doing (say, by computing population firing rate) and then play with
parameters until you find parameter sets that give you a couple of different network behaviors.
Remember there can be heterogeneity between cells, and the sign of c determines the type of
connection.
B) What would you need to do to introduce a particular connectivity pattern, say, if all cells
are lying on a line and there is ”mexican-hat” connectivity? (You don’t have to implement this
network)
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3. a) Consider tonic bursting and tonic spiking varieties of the Izhikevich model from this
website: http://www.izhikevich.org/publications/figure1.m.
What was changed to switch from one to the other? Explain from dynamics perspective why
that change had the desired effect?
b) Do the same thing comparing tonic spiking and phasic spiking versions.
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