Computational Neuroscience
aka MATH 378/478, BIOL 378/478, EECS 478, NEUR 478, EBME 478, COGS 378
Fourth Homework. Due 2010/03/01 in class.
Questions with a (*) should be completed by students enrolled in any 478 cross-listing (i.e. for
graduate credit). Questions marked (**) should be completed by students enrolled in MATH 478.
You are encouraged to work with other members of the class to solve the problems, however, the
work you turn in must be your own. For full credit, please show enough intermediate steps in your
calculations for the instructor to have confidence that you understand the solution method you
employed.
1. (Spikes, Decisions and Actions Exercise 9.8.1): Use the Poincaré-Bendixson theorem to prove
the existence of limit cycles for Wilson’s equation 9.7:
C dv/dt = (−g1 (v)(v − E1 ) − g2 r(v − E2 ) + Iapp )
dr/dt = (−r + 1.35v + 1.03)/τr
g1 (v) = 17.81 + 47.71v + 32.63v 2
τr = 1.9, c = 0.8, E1 = 0.55, g2 = 26, E2 = −0.92
with Iapp = 0.5. Determine the stability of the steady state and then construct an appropriate
closed curve in the phase plane through which all trajectories must pass to complete the proof.
Show an accurate graph of your construction. (You may use Matlab, XPP, or a similar tool
if you wish.)
2. (Spikes, Decisions and Actions Exercise 9.8.2): When the K+ concentration outside a neuron
gets too high, the equilibrium potential EK becomes less negative. Prove that the resting
state for Iapp = 0 of Wilson’s equation 9.7 (see previous problem) can become unstable under
these conditions for a range of EK values. Simulate this situation to show that spike trains
are generated. Identify the type of bifurcation that leads to limit cycles. What happens if
EK continues to increase?
3. (a) Use XPP/AUTO or another system (e.g. Matlab’s numerical continuation package
MATCONT) to create a bifurcation diagram and superimposed trajectory for a canonical
elliptic burster, written in normal form as:
u0 = u(λ + R − R2 ) − v + c
v 0 = v(λ + R − R2 ) + u + c
λ0 = (R0 − R)
R = u2 + v 2
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with parameters = 0.01, R0 = 0.3, c = 0.01. Let time run from 0 to 1000. See the instructions on pages 193-4 of Simulating, Analyzing, and Animating Dynamical Systems
by Ermentrout (Exercise 7.5.1 of this book).
(b) (**) Discuss the classification of the parabolic burster in part (a) in terms of the sixteen
types of “generic planar bursters”. (See also problem 6.)
4. Classification of Single Cell Dynamics.
(a) Hodgkin described three types of cell responses to a step current. Type 1 cells exhibit
periodic firing with a frequency that depends on the strength of the injected current. In
particular, type 1 cells can be driven at quite low frequencies by having injected currents
just above the rheobase (minimal current to induce spiking). Type 2 cells exhibit periodic
firing with a frequency that does not depend strongly on the strength of the injected
current. These cells show a clear minimum firing rate. Type 3 cells spike once upon
stimulation with a step current and then stop spiking.
From a dynamical systems point of view, onset of spiking is related to a bifurcation:
loss of stability of a stable fixed point. There are four generic (or “codimension one”)
bifurcations of equilibria for smooth systems of differential equations depending on a
single parameter (the injected current, for example): subcritical Hopf bifurcation, supercritical Hopf bifurcation, saddle-node bifurcation, and saddle-node-on-invariant-circle
(SNIC) bifurcation.
Explain the relationships, if any, between Hodgkin’s classification of excitability and the
different codimension one bifurcations of equilibria.
(b) (*) Supplement your answer to part (a) by explaining in detail the role of quasi-separatrixcrossing in excitability of class three neurons.
5. Visit the ModelDB site (http://senselab.med.yale.edu/modeldb) and find the .ode files associated with the paper
Prescott SA, De Koninck Y, Sejnowski TJ (2008) Biophysical basis for three
distinct dynamical mechanisms of action potential initiation. PLoS Comput Biol
4:e1000198
by searching under “Prescott” (or accession number 116123). Download Prescott et al. ’s
version of the Morris-Lecar model, 2d-ML.ode, and run it in XPP.
(a) Set the parameters to values for which modifying the applied current I =i dc through
the range 0 ≤ I ≤ 80 leads to a SNIC bifurcation. Superimpose nullclines drawn for
I = 0, 40, 80. Clearly label the nullclines for the different current values (use (TT) for
Text/Text etc.), and label the key parameter (βw ) as well. Take a single screen shot
of the (v, w) phase plane showing the above elements as well as a sample trajectory
illustrating the limit cycle when I = 80.
(b) Repeat the steps for (a) above, for values of the parameters for which varying I leads to
a subcritical Hopf bifurcation.
(c) Repeat the steps for (a) above, for values of the parameters for which a sudden change
in the input current from 0 to 80 causes a single spike. Plot a trajectory of such a spike
in the (v, w) phase plane, starting from the stable steady state present when I = 0, and
plotting the trajectory from that initial condition when the current is changed to I = 80.
You may omit the nullclines for I = 40 for clarity.
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6. (**) Classification of bursting dynamics.
There are four codimension-one planar bifurcations of limit cycles. Together with the four
codimension-one bifurcations of equilibria, there are sixteen dynamically distinct “types” of
planar bursting (“planar” meaning that the fast subsystem is two dimensional). Another
type of intermittent firing pattern can come from so-called “circle-circle” bursting, in which
the “resting state” is actually a small amplitude limit cycle. Classify all planar circle-circle
bursters.
Hint: consult Izhikevich, Dynamical Systems in Neuroscience, Chapter 9.
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