Winter 2016 PHYS 178/278, Homework 4
Due midnight on March. 12th
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Cable Equation
Real axons and dendrites are essentially long thin cylinders (Fig. 1). The voltage inside a axon/dendrite changes
with distance (x) so there is a longitudinal current flowing down to the axon. Meanwhile, the current can leak
through the membrane that gives an additional radial trans-membrane current. Using the circuit diagram
shown in Fig. 1, derive the the cable equation
∂V (x, t)
∂ 2 V (x, t)
+ V (x, t) − λ2
= 0.
∂t
∂x2
Let the membrane impedance consists of a parallel combination of a capacitance cm = m /L, a linear resistance
q
rm = ρm L, where L is the thickness of the membrane; the resistivity inside the axon ρc ; the constant λ = ρρmc aL
2 ,
and τ = rm cm .
Hint:
1) What are the currents that flow into and out of node x? Using Kirchhoff’s junction rule (conservation of current),
write down an equation for this.
2) Get each current (longitudinal and radial) for a segment length ∆x based on the circuit diagram.
3) Plug all the currents into the equation you got in step 1). Now you have a circuit equations for a segment of
length ∆x.
4) The exact cable equation can be obtained by taking the limit ∆x → 0.
τ
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Single Layer Perceptron
Here you will demonstrate how a perceptron network performs a NAND (NOT-AND) function. A simple perceptron
network consists of two inputs x1,2 and an output d (x1 , x2 ).
x1
0
0
1
1
x2
0
1
0
1
d (x1 , x2 )
1
1
1
0
Table 1: NAND truth table
The output y is defined as:
(
1 w1 x1 + w2 x2 + b > 0
y=
0 otherwise
(1)
To learn the function, the weights have to be updated according to:
wi (t + 1)
=
wi (t) + α [d (x1 , x2 ) − y(t)] xi
(2)
b(t + 1)
=
b(t) + α [d (x1 , x2 ) − y(t)]
(3)
α is the learning rate. The value of w1 , w2 and b that perform the NAND function will be learned when you
repeatedly introduce the four truth statements as inputs.
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Figure 1: Equivalent electrical circuit for a cylindrically symmetric segment of axon/dendrite.
1. Run the perceptron simulation by constructing a “test set”. What are the resulting w1 , w2 and b?
2. Does your result correctly separate the 2-d input space?
3. What α did you use? Vary α and plot the number of iterations as a function of α.
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