Winter 2016 PHYS 178/278, Homework 2
Due midnight on Feb. 6th
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Random walks
Random walks can be a good model of noise in neurons. A random walk in one dimension is the recursion relation:
(
+1, pR = 12
xt+1 = xt +
−1, pL = 21
where pR and pL are the probabilities to move to the right or left in each time step.
1.1 Generate a one dimensional random walk starting from x0 = 0 for T = 100 time steps. Plot xt as a function of t.
1.2 Generate 1000 random walks for T = 100, 500, 2000. Plot hxT i as a function of T , where h·i means averaging
over the 1000 repetitions for each value of T . Does hxT i depend on T ?
1.3 For the same random walks from 1.2, plot x2T as a function of T . How does x2T depend on T ?
1.4 Use the central limit theorem to derive an expression for x2T as a function of T and plot it along with the
simulations from 1.3. Does your expression match the simulations?
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Conductance models and ion channels
2.1 The attached MATLAB file, HHModel.m, includes the original equations written by Hodgkin and Huxley:1
C
dV
dt
dn
dt
dm
dt
dh
dt
= −g L (V − VL ) − g N a m3 h (V − VN a ) − g K n4 (V − Vk ) + I (t) ,
(1)
=
(1 − n) αn − nβn ,
(2)
=
(1 − m) αm − mβm ,
(3)
=
(1 − h) αh − hβh ,
(4)
and all the α and β are given by,
αn (V )
=
αm (V )
=
αh (V )
=
0.01 (−V + 10)
−V
,
β
(V
)
=
0.125
exp
;
n
−V +10
80
e( 10 ) − 1
0.1 (−V + 25)
−V
,
β
(V
)
=
4
exp
;
m
−V +25
18
e( 10 ) − 1
−V
1
0.07 exp
, βh (V ) = −V +30
,
(
20
e 10 ) + 1
1 The only difference from the original paper is that we use the modern sign convention for the voltage. Notice that this original
formulation is in terms of a “maximal conductance” for each type of “current,” while in modern language we could talk about the
number of each type of channel. In fact, the more phenomenological description persists, because it corresponds more directly to what
is measured, but this allows us to forget that parameters such as g K actually measure the number of copies of a protein that have been
inserted into the membrane.
A. L. Hodgkin & A. F. Huxley, “A quantitative description of membrane current and its application to conduction and excitation in
nerve.” The Journal of Physiology 117 (4): 500–544 (1952).
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where N a and K refer to sodium and potassium channels, respectively; time is measured in milliseconds and voltage
V is measured in milliVolts (mV). These equations are intended to describe a small patch of the membrane, and so
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2
2
2
many parameters are given per unit area: C = 1µF/cm , g L = 0.3 mS/cm , g N a = 120 mS/cm , g K = 36 mS/cm 2 ;
the reversal potentials are VL = 10.613 mV, VN a = 115 mV, VK = −12 mV.
Rewrite Eqn. (2), (3) and (4) in terms of equilibrium values and relaxation times for the gating
variables, e.g.,
1
dm
=−
[m − meq (V )] ,
dt
τm (V )
where meq (V ) is the solution when system reaches dynamical equilibrium, and τm (V ) is the relaxation time. Both meq (V ) and τm (V ) can be expressed in terms of αm (V ) and βm (V ). Plot these
quantities: meq (V ), neq (V ), heq (V ), and τm (V ), τn (V ), τh (V ), as a function of voltage V . Can you
explain, intuitively, the form of the curves?
2.2 Use the MATLAB files, HHModel.m and plotHH.m, to simulate a Hodgkin-Huxley neuron in response to different constant current inputs. What is the minimal input (threshold current) required for spike generation?
2.3 Explore the frequency of the pulses as a function of current. Count the number of spikes in a window
of fixed size (say, 1000 ms) for different values of the driving current and plot the “F-I curve” (the frequency as a
function of the driving current). Can the cell be driven to fire at an arbitrarily high frequency?
Hint: the MATLAB function findpeaks(...) could be helpful for counting the number of spikes. You may need to
write a short MATLAB code to finish this plot.
2.4 The leaky integrate-and-fire neuron is a simplified of spiking neuron model. The equation for a dimensionless
leaky integrate-and-fire neuron is:
(
v̇ = −v + I if v < vth
(5)
reset v to 0 if v ≥ vth , vth = 1
Here is how it works: if a large enough input current is applied, the membrane voltage increases with time until
it reaches a constant threshold vth , at which a spike occurs by stipulation and the voltage is reset to its resting
potential, after which the model continues to run. Find the F-I curve without simulating the neuron model
and plot it alongside the F-I curve you found in 2.3. Are they qualitatively the same? Why or why
not?
Hint:
To find the F-I curve solve the differential equation assuming there is no reset condition.
Then set the initial condition to v(t = 0) = 0.
Find the time it takes the neuron to spike from that initial condition, as a function of I.
2.5 From dimensional analysis and using the electrophysiological properties of “typical cells” discussed in class,
estimate the number of ions that cross the membrane during a single action potential.
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More fun with models of spiking neuron
Play with MATLAB file rizhikevich.m3 (including izhikevich.jpg in the same folder in order to run the file) and
generate theses 8 spiking patterns: (F), (C), (G), (I), (M), (O), (P) and (Q) shown in Fig. 1. Briefly describe
their spiking features and the similarity/difference compared to the Hodgkin and Huxley neuron in Prob. 2.2.
2 The
3 The
symbol S (siemens) is the unit of electric conductance in SI unit; S(siemens) = 1/Ω(Ohm).
MATLAB file and the figure is from http://www.izhikevich.org/
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(A) tonic spiking
(B) phasic spiking
(C) tonic bursting
(D) phasic bursting
(E) mixed mode
(F) spike freq. adapt
(G) Class 1 excitable
(H) Class 2 excitable
(I) spike latency
(J) subthreshold osc.
(K) resonator
(L) integrator
(M) rebound spike
(N) rebound burst
(O) thresh. variability
(P) bistability
(R) accomodation
(S) inh. induced sp.
(T) inh. induced brst.
(Q) DAP
Figure 1: Summary of the neuro-computational properties of biological spiking neurons. Each horizontal bar
denotes a 20-ms time interval. (Electronic version of the figure and reproduction permissions are freely available at
www.izhikevich.com).
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