1 R.V k V k 1 / I.k/ here; we ll stimulate the action potential another way.) Note that this further simplifies to. m 3 k h k.

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1 1. The goal of this problem is to simulate a propagating action potential for the Hodgkin-Huxley model and to determine the propagation speed. From the class notes, the discrete version (i.e., after breaking it up into compartments of length x and radius a) of the cable equation in the constant diameter case is C m dv k dt D 1 R.V kc1 V k / 1 R.V k V k 1 / I.k/ m ; where R D r L x=a 2, C m D 2axc m, I m.n/ D 2axi m.x n /. (We won t use any electrode current here; we ll stimulate the action potential another way.) Note that this further simplifies to dv k dt D a 2r L c m x 2.V kc1 2V k C V k 1 / i m.x k / c m As before, i m.x k / D g L.V k E L / C g Na m 3 k h k.v k E Na / C g K n 4 k.v k E K / with g L D 03 ms/cm 2, g Na D 120 ms/cm 2, g K D 36 ms/cm 2, E L D mv, E Na D 50 mv, and E K D 77 mv. The gating variables m k, h k and n k (note we need one set of them for each compartment, hence the subscript k) are determined by equations of the form dz dt D z.v /.1 z/ ˇz.V /z ; where z denotes one of them. For the gating variables m k, h k, and n k use the rate functions m.v / D 01.V C 40/ 1 exp. 01.V C 40// ; ˇm.V / D 4 exp Œ.V C 65/=18 h.v / D 007 exp Œ 005.V C 65/ ; n.v / D ˇh.V / D 10 1 C exp Œ 01.V C 35/ 001.V C 55/ 1 exp Œ 01.V C 55/ ; ˇn.V / D 0125 exp Œ V C 65/ In addition, use c m D 1 F/cm 2 and for the squid axon take r L D 354 -cm. (Note these units are not all compatible with one another; some conversion of units will be necessary.) One way to simulate the propagating action potential is to use Matlab s ode15s. This is a solver for stiff sets of equations, i.e., ones with wide ranges of time scales. (If you know another method that will work to solve these equations, you are free to use it.) The solver ode15s works like ode45, so you will a function that takes as input the time and a big column vector comprised of N voltages V k, followed by N gating variables m k, then N gating variables h k and finally N gating variables n k. (So the overall vector has 4N components.) That function will need to return a big vector, the first N components of which are dv k =dt, followed by the N components dm k =dt, then the N components dh k =dt and finally the N components dn k =dt. The equations for the gating variables can be done using single formulas, if one uses Matlab s dot notation to operate on vector elements term-by-term, e.g., alpha_m = 0.1(V+35)./(1-exp(-0.1(V+35))).

2 The only other issue is how to handle the cable part of the equation, the term D.V kc1 2V k C V k 1 /, where D D a=.2r L c m x 2 /. For this, note that if we make a big tri-diagonal matrix 2 A D D then D.V kc1 2V k C V k 1 / D A EV, where EV is the column vector with components V 1, V 2,... V N ; thus, this term can be determined with simple matrix multiplication. Also, since most of the matrix elements are zero, it saves memory to represent it as a low-storage sparse matrix (where only the non-zero terms are stored). If one makes a vector of all ones using e = ones(n,1); then the matrix A can be constructed using the Matlab code A = D*spdiags([e -2e e],[-11],n,n); One final thing if one looks at the k D 1 term in the product A EV, the V 0 term is missing (and similarly for V N C1 ). This is equivalent to saying V D 0 at the ends of the x interval. This can cause problems, and it s better to assume D 0. One can do this relatively easily by instead assuming that V 0 D V 2, and similarly that V N C1 D V N 1. (Note if V 0 D V 2 then EV is symmetric about k D 1.) One can then implement this by modifying A after it s been created with spdiags using the lines A(1,2)=A(1,2)*2; and A(N,N-1)=A(N,N-1)*2;. Instead, then, one will have the matrix A D D This completes the basic instructions to set up your Matlab program. Here s the explicit part about what to do Run your code using a axon radius of cm (i.e., 238 m), a spacing x D 025 cm, and N D 201 points (so, the total length is 50 cm). Use an initial voltage of -65 mv, an initial m value of 0.05, an initial h value of 0.6 and an initial n value of 0.3, except stimulate the left-most 1 cm of the axon by setting the initial voltage in that section to 0 mv. This should be sufficient to generate a propagating action potential. You should plot the shape of the propagating action potential at the middle point of the region, and determine the action potential s velocity. One way to do the latter is to use an event function that records the times at which different sections of the cable (say, at 25 cm and 40 cm) first reach 0 mv. (You can stop the simulation when the second event happens.) The velocity, of course, will be the distance traveled divided by the time interval the event function gives

3 The ion channel models we are using come from Hodgkin and Huxley s original paper. 1 These models were fit to data measured at 63 ı C. In their paper, they also report the propagation velocity at 18.5 ı C. In addition to showing the shape and determining the propagation velocity at 63 ı C, also do this at 18.5 ı C. To change the model to work at different temperatures, all of the ion channel rates (i.e., the s and the ˇs) must be multiplied by a temperature-dependent factor the ion channels speed up at higher temperatures..t 63/=10 The appropriate factor is Q where T is the temperature and Q D 3. (In other words, it was determined experimentally that the ion channels speed up by a factor of 3 for every 10 ı C.) Using an appropriately modified version of your code, plot the propagating action potential shape and determine its velocity at 18.5 ı C. How well do your results agree with Hodgkin and Huxley s prediction of 18.8 m/s? (Or their experimentally determined velocity of 21.2 m/s?) 2. (This is a problem due to Dayan and Abbott.) A Hopfield associated memory has firing rates or activities for a set of networked neurons, v k for k D 1, 2,..., N (or collectively, a vector of activities v), that take values of either C1 or 1, and are updated at discrete time steps according to the rule 0 1 NX v i.tc1/ D M ij v j.t/ A ; (1) where j D1 8 < C1 if z 0 sgn.z/ D 1 if z < 0 The matrix M of synaptic weights is constructed from P memory vectors v m (m D 1, 2,..., P ) that are also composed of elements that are either C1 or 1. M is determined through the sum of outer products M ij D.1 ı ij / PX v m i v m j (2) Here ı ij D 1 if i D j and 0 otherwise, and this makes the diagonal elements of M zero. Consider a 100-element network (N D 100) and construct P memory states by randomly assigning +1 and -1 values with equal probabilities to the N elements of each v m. Using these memory vectors, set the matrix of synaptic weights according to equation (2). Then, study the behavior of the network by iterating equation (1). To measure how close the state of the network at time t, v.t/, is to a particular memory state, say v 1, define the overlap function q.t/ D.v 1 v.t//=n. Note this will be equal to 1 if v.t/ D v 1, is near zero if v.t/ is unrelated to v 1, and is equal to -1 if v.t/ D v 1. Set the initial state v.0/ so that it has a positive overlap, q.0/ D q 0 > 0, with memory state v 1, and determine the probability that this memory state is recovered perfectly as a function of the number P of stored memory vectors. Do this by computing the final overlap q.final/ (when the memory state stops changing) for a number of different random sets of memory vectors v m (m D 1, 2,..., P ) and different 1 A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117 (1952) pp md1

4 random initial vectors v.0/ with q.0/ D q 0 D 09 always the same value. The probability of perfect recall will be the fraction of times q.final/ is equal to 1. The goal of this assignment Your task is to produce a plot of the estimated probability that q.final/ D 1 as a function of P, for P between 1 and 30. A plot of the probability of perfect recall is a way to assess the storage capacity of the network. Here are some hints that will help you do this with Matlab (a) You can generate a matrix whose P columns are the memory vectors with a single Matlab statement V=2*randi(2,N,P)-3;. The randi generates a random matrix of integers between 1 and 2 (i.e., either 1 or 2) of size N xp, multiplying by 2 gives either 2 or 4, and subtracting 3 gives -1 or 1. [Hint try this by typing it in the command window without the semicolon with, say P=2 and N=5, to see what it is doing. You can run it several times to make sure you always get patterns of +1 and -1 that are different each time.] (b) If V is the matrix whose columns are the P memory vectors, then the Matlab statements M=V*V, followed by M=M-diag(diag(M)) gives the synaptic connectivity matrix. The Matlab command diag extracts the diagonal of a matrix as a vector, or turns a vector into a diagonal matrix. [Hint you can run this by hand using the smaller V you generated above to make sure you understand how this removes the diagonal terms from the matrix.] (c) With a random synaptic connectivity matrix and a set of random initial vectors the results will vary from trial to trial. It is therefore helpful to repeat the iteration with a number of initial random vectors, and also to use a number of different random sets of memory vectors making up the matrix M. The parallel capabilities of Matlab can help with some of this if one constructs v so it has K different columns, one can update all K columns simultaneously using the same iteration as described above. A way to make K copies of v 1 in Matlab is with the command v=repmat(v(,1),1,k);. Here repmat is the command to replicate and tile an array. [Hint Again, you can try this by hand to understand what it does.] (d) The Matlab sgn function isn t the same as the one above in the iteration given by Equation (1), since sgn(0)=0. One way to fix this is to do vnew=sgn(m*v); zero_entries=find(vnew==0); vnew(zero_entries)=1; [Hint The Matlab find function is very useful for working with selected elements of vectors and matrices. It is worth spending some time to understand how it works.] (e) The random initial vectors should all have the same q 0. One way to do this is the have L elements or components of the initial v different from v 1 and N L elements the same. For each component that is different, q.t/ D.v 1 v.t//=n will decrease by 2=N. (When a component is the same, the product of that component with the same component in v 1 will give C1, and when it is different it gives 1.) You can randomly flip L elements of each column of v with the code snippet

5 for j=1k flip=randperm(n,l); v(flip,j)=-v(flip,j); end Here randperm(n,l) selects random set of L N numbers from 1; 2; N without replacement. (There might be some way to do this without a for loop, but this way is easy to understand.) Pick L so that the initial q.0/ D 09. [Hint you can also type these lines into the command window to understand them. Matlab will not execute the commands inside the for loop until you type the end.] (f) The above iteration, equation (1), has been written as a parallel update. This speeds up the simulation a lot compared with one-at-a-time asynchronous updates, but the iteration with parallel updates does not always converge sometimes one gets oscillations with a period of 2, i.e., the iteration goes back and forth between two different vectors. As a result, you should stop the iteration if the iteration two steps ago is the same as the new one. (Note this also works if one gets to a steady state where there is no oscillation.) One way to do this is to have three variables vold, v and vnew, and after each iteration check to see if vnew is identical to vold; One checks to see if vectors are identical in Matlab with the test isequal(vnew,vold). If the test is true one stops the iteration, otherwise one sets vold=v and v=vnew (in that order you should understand why the order is important) and then continues the iteration. It s also always a good idea to have a limit on the total number of iterations, so that if something doesn t work as expected your program doesn t get stuck in an infinite loop. A way to do this is using a for loop with a maximum number of iterations. In this case, the way to exit the loop if the test is true is with a Matlab break statement. (g) For each value of P (the number of stored vectors) you want to calculate q=v *V(,1)/N and determine the fraction of the q.final/ values that are equal to 1. This is an estimate of the probability that q.final/ D 1. Make sure you use a sufficient number of different sets of random memory vectors and number K of randomly permuted initial vectors so that the resulting plot of the estimated probability as a function of P is relatively smooth. (h) How many memory vectors can be stored when N D 100 so that one has at least a 90% chance of perfect recall? This is one measure of the storage capacity of a Hopfield network of 100 neurons.