### Final Project Discussion¶

Is it possible to condense a large recurrent network into a cannonical set of nonlinear responses. I took to approaches to this. One was to use PCA to identify the similarity between neurons and then merge based on this criterion. Because I was not able to merge more than a dozen or so neurons in this way I then tried merging these neurons and iteratively training. The thinking behind this was that if I could merge several neurons and still "lie close" to a solution that training would take me towards the nearest solution. Next I asked how the types of solutions that a network can converge to depend on the size of the network.

what's the background and prior art?

I didn't find any work condensing trained RNNs to small cannonical sets. Presumably this is because the bulk of RNN work is done in areas where we want large networks with lots of expressive power. The goal of this project was to see if there is a relatively simple way to reduce RNNs to smaller networks where it is more tractable to study the dynamics of the network in detail.

what sources did you use?

I used a library private library for training the reccurent networks using conjugate gradient and hessian free methods that was developed by a post-doc in my lab as well as the behavioral analog developed in the lab for prior work.

what did you do?

I used PCA to categorize neurons into different groups and then tried to merge neurons using differnent critera (this involves essentially passing weights from one neuron to another and deleting all of the weights from that neuron). Next I tried to perform this iteratively, training the network 100 times after each merging step and then

what worked, what didn't?

I wasnt able to condense more than a dozen or so neurons using PCA to choose, pairs. I also tried using correlation but obtained similar resuts. Next I tried to perform this process iteratively but I ended up with a ramping solution which is not similar to the network I started with. I looked a bit at how the size of the network affects not only the possible expressive power but also constrains the type of solutions the network can make

how did you evaluate progress?

This was relatively easy because I wanted to obtain a network that could still do the task which I could assess simply by running network forward. I also looked at the PC's of the network and asked how many were required to

what are the implications for future work?

Perhaps there is something interesting related to the types of solutions found by RNNs that can be studied using a very simple task like this one.

SEMESTER PROJECT IDEAS:

The main thing that I want to get out of this project is to review some of the different ways that people use neural networks, not for machine learning but for modeling real neurons. To this end I started by doing several small projects making different "toy" networks getting an idea about how to do simple simulations of spiking neurons, small rate based recurrent networks, ring attractors etc. After doing this I turned to a larger reccurent network that was trained through hessian free back-propegation through time to preform an analog of a behavioral task that was developed and is used in a lab that I spend time in this past semester.

Initial Idea: Make a biophysical model of a syrinx and get it to produce vocalizations Simulate and oscillating membrane and manipulate the pressure in the organ to output different sound pressures that the ‘beak’

(https://en.wikipedia.org/wiki/Syrinx_(bird_anatomy)) The problem with this idea is that there seems to be a lot of work already done on this and I don’t see a clear way that I would like to build on what has already been done. For instance there is a lab in Argentina which basically works solely on my project idea (http://www.lsd.df.uba.ar/)

### WEEK 1:¶

This week I wanted to get some intuition about about how to simulate spiking neurons so I undertook the relatively simple task of putting together a simulation of a single spiking neuron using a description of the differential equation that descibes the ionic currents

In [13]:
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np

########## PARAMETERS ########## (see Tesileanu et al.,)
n = 300 #number of neurons
Vrest = -72.3 #reset potential in mV
Vthr = -48.6 #threshold in mV
tMEM = 24.5 #membrane time constant in ms
ref = 10 #refractory period in ms
tAMP = 6.3 #ampa time constant in ms
tNMDA = 81.5 #nmda time constant in ms
tINH = 20 #inhibitory time constant in ms
R = 353 #MegaOhms input resistance
$$\text{the differential equation governing the voltage of neuron i is given by} \\ \tau_{m} \dot{V_i} = V_{r}-V_i + R(I_i^{AMPA}+I_i^{NMDA}) - V_{inh}$$$$\tau_{m} \quad \text{is the time constant of the membrane}\\$$$$V_{r} \quad \text{is the resting potential of the neuron}\\$$$$I_i^x \quad \text{current input of type x into neuron i at time t}\\$$$$V_i \quad \text{is the voltage of neuron i at time t}\\$$$$R \quad \text{is the input resistance}\\$$
In [9]:
########## TEST CASE ########## (using random current and inhibitory input)
tstep = 5000
V = np.zeros(tstep)
V[0] = Vrest
I = (np.random.rand(tstep)-0.3)
inh = np.random.rand(tstep)*100
#inh = 35*np.ones(tstep)
j = 0

spike = False

for i in range (1,tstep):
if V[i-1] > Vthr:
spike = True

if j == ref:
spike = False

if spike == True:
spike = True
if j < 1:
V[i] = 50 + np.random.rand(1)*25
else:
V[i] = Vrest-20+j + np.random.rand(1)*25
j+=1

if spike == False:
j = 0
dV = (1/tMEM)*(Vrest - V[i-1] + R*(I[i-1]) - inh[i-1])
V[i] = V[i-1]+dV

fig =plt.figure(figsize=(6, 3), dpi= 180, facecolor='w', edgecolor='k')
plt.plot(V[0:500],color = 'k', linewidth = 0.5)
plt.plot(np.ones(500)*Vthr,'--r', linewidth = 1)
plt.plot(np.ones(500)*Vrest,'--b', linewidth = 1)

fig =plt.figure(figsize=(6, 3), dpi= 180, facecolor='w', edgecolor='k')
plt.plot(V,color = 'k', linewidth = 0.5)
plt.plot(np.ones(tstep)*Vthr,'--r', linewidth = 1)
plt.plot(np.ones(tstep)*Vrest,'--b', linewidth = 1)
Out[9]:
[<matplotlib.lines.Line2D at 0x110c9a320>]

#### Other possible ways?¶

$$\text{the differential equation governing the voltage of neuron i in this model is given by} \\ {V_i} = V_{i-1} + e^{\frac{R*(I_i^{AMPA}+I_i^{NMDA})}{\tau_m}} - V_{inh}$$$$\tau_{m} \quad \text{is the time constant of the membrane}\\$$$$V_{r} \quad \text{is the resting potential of the neuron}\\$$$$I_i^x \quad \text{current input of type x into neuron i at time t}\\$$$$V_i \quad \text{is the voltage of neuron i at time t}\\$$$$R \quad \text{is the input resistance}\\$$

### WEEK 2:¶

This week I put together a simple two neuron recurrent network as shown below and simply tested different inputs into this network. It has a diagonal weight matrix and thus input along the eigenvectors excite of the of the two modes of the network as shwon in the animation

In [12]:
display(HTML("<table><tr><td><img src='F2.large.jpg'></td><td><img src=''></td></tr></table>"))

### Gaining intuition for simple recurrence in small networks¶

In [14]:
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
from IPython.display import HTML
from numpy import linalg as LA
import matplotlib.animation as animation

## SET PARAMETERS ##
tstep = 800
M = np.array([(0,0.8),(0.8,0)])
v = np.zeros([2,tstep])
h = np.zeros([2,tstep])
val,vec = LA.eig(M)
h[0,30:40] = np.ones([1,10])
h[1,130:140] = np.ones([1,10])
h[:,230:240] = np.ones([2,10])
h[0,330:340] = -np.ones([1,10])
h[1,430:440] = -np.ones([1,10])
h[:,530:540] = -np.ones([2,10])
h[:,630:640] = np.ones([2,10])
h[0,630:640] = h[0,630:640]*vec[0,1]
h[1,630:640] = h[1,630:640]*vec[1,1]
h[:,730:740] = np.random.rand(2,1)*np.ones([2,10])
tao = 4

## RUN DYNAMICS FORWARD ##
for i in range(1,tstep):
dv = (1/tao)*(-v[:,i-1] + np.dot(M,v[:,i-1]) + h[:,i-1]);
v[:,i] = v[:,i-1]+dv;
In [15]:
# First set up the figure, the axis, and the plot element we want to animate
fig = plt.figure(figsize=(5, 3), dpi= 180, facecolor='w', edgecolor='k')
#ax = plt.axes(xlim=(-3, 3), ylim=(-3, 3))

line, = ax1.plot([], [], lw=1, color = 'k',)
line2, = ax1.plot([], [], 'o',lw=1, color = 'k',markersize=5)

line3, = ax2.plot([], [], '-',lw=3, color = 'k')
ax1.grid()
ax2.grid()

# initialization function: plot the background of each frame
def init():
line.set_data([], [])
line2.set_data([], [])
line3.set_data([], [])
return line, line2, line3

# animation function.  This is called sequentially
def animate(i):
x = v[0,1:i+1]
y = v[1,1:i+1]
line.set_data(x, y)
line2.set_data(v[0,i+1], v[1,i+1])
line3.set_data([0,h[0,i+1]], [0,h[1,i+1]])
return line,

# call the animator.  blit=True means only re-draw the parts that have changed.
anim = animation.FuncAnimation(fig, animate, init_func=init,
frames=799, interval=20, blit=True)

ax1.plot([0,vec[0,0]],[0,vec[1,0]],'r-',linewidth = 0.5)
ax1.plot([0,vec[0,1]],[0,vec[1,1]],'r-',linewidth = 0.5,label='eigenvectors')

handles, labels = ax1.get_legend_handles_labels()
ax1.legend(handles, labels)

ax1.set_title("network activity")
ax2.set_title("network input")
plt.tight_layout()
plt.close()
HTML(anim.to_html5_video())
Out[15]:

### WEEK 3:¶

Reproduce respiratory model from Rubin et al., 2009

https://en.wikipedia.org/wiki/Pre-B%C3%B6tzinger_complex

In [174]:
#%%
import matplotlib.pyplot as plt
import numpy as np
from numpy import linalg as LA
#%% parameters

# Conductances:
C=20
g_Nap=0
g_k=5.0
g_L=2.8
g_synE=10.0
g_synI=60

# Reversal potentials:
E_Na=50
E_k=-85
E_L=-60
E_synE=0
E_synI=-75

# weights
a_12=0.4
b = np.array([(0,0,0.3,0.2),(0,0,0.05,0.35),(0,0.25,0,0.1),(0,0.35,0.35,0)])
c = np.array([(0.115,0.07,0.025),(0.3,0.3,0),(0.63,0,0),(0.33,0.4,0)])

V_half=-30
kV = np.array([8,4,4,4]) #output slopes
tau_naMax=6e3
d = np.ones(3) #external inputs

#%%
# lambda functions
f = lambda x, y: 1/(1+np.exp(-(x-V1_2)/y))
hInfNaP= lambda u: 1./(1+np.exp((u+48)/6))
mNaP= lambda u: 1./(1+np.exp(-(u+40)/6))
TauhNaP= lambda u,tauMax: tauMax/np.cosh((u+48)/12)
mK= lambda u: 1./(1+np.exp(-(u+29)/4))
f= lambda u, kv: 1./(1+np.exp(-(u-V_half)/kv))

#%%
# Initialize variabiles
numsteps=100000
dt=1e-1
h_NaP=np.zeros(numsteps)
I_K=np.zeros(numsteps)
I_NaP=np.zeros(numsteps)
V = np.zeros((4,numsteps))
V[:,0] = np.array([-50.82041612, -36.22710468, -36.83931196, -41.83978807])
I_L = np.zeros((4,numsteps))
I_synE = np.zeros((4,numsteps))
I_synI = np.zeros((4,numsteps))

#%%
for j in range(0,numsteps-1): #TIME LOOP
# updating parameters
tau_hNaP=TauhNaP(V[0,j],tau_naMax)
h_NaP[j+1]=dt*1/tau_hNaP*(-h_NaP[j]+hInfNaP(V[0,j])+h_NaP[j])
I_K[j+1]=g_k*(mK(V[0,j])**4)*(V[0,j]-E_k)
I_NaP[j+1]=g_Nap*mNaP(V[0,j])*h_NaP[j+1]*(V[0,j]-E_Na)
# get inputs
inhibitory_activation = np.dot(b,f(V[:,j],kV))
excitatory_activation = np.dot(c,d)

for i in range(0,4): # NEURON LOOP
I_L[i,j]=g_L*(V[i,j]-E_L)
if i == 1:
I_synE[i,j]=g_synE*(V[i,j]-E_synE)*(a_12*f(V[0,j],kV[0])+excitatory_activation[i])
else:
I_synE[i,j]=g_synE*(V[i,j]-E_synE)*excitatory_activation[i]
I_synI[i,j]=g_synI*(V[i,j]-E_synI)*inhibitory_activation[i]
if i == 0:
V[i,j+1]=dt*1/C*(-I_NaP[j]-I_K[j]-I_L[i,j]-I_synE[i,j]-I_synI[i,j])+V[i,j]
else:
In [175]:
fig = plt.figure(figsize=(4, 5), dpi= 180, facecolor='w', edgecolor='k')

plt.subplot(4, 1, 1)
plt.plot(np.arange(0,numsteps)*dt/600,f(V[0,:],kV[0]),'k',linewidth = 2)
plt.title('neuron 1')
plt.subplot(4, 1, 2)
plt.plot(np.arange(0,numsteps)*dt/600,f(V[1,:],kV[1]),'r',linewidth = 2)
plt.title('neuron 2')
plt.subplot(4, 1, 3)
plt.plot(np.arange(0,numsteps)*dt/600,f(V[2,:],kV[2]),'b',linewidth = 2)
plt.title('neuron 3')
plt.subplot(4, 1, 4)
plt.plot(np.arange(0,numsteps)*dt/600,f(V[3,:],kV[3]),'orange',linewidth = 2)
plt.title('neuron 4')
plt.show()

fig = plt.figure(figsize=(4, 2), dpi= 180, facecolor='w', edgecolor='k')
plt.plot(np.arange(0,numsteps)*dt/600,f(V[0,:],kV[0]),'k',linewidth = 1)
plt.plot(np.arange(0,numsteps)*dt/600,f(V[1,:],kV[1]),'r',linewidth = 1)
plt.plot(np.arange(0,numsteps)*dt/600,f(V[2,:],kV[2]),'b',linewidth = 1)
plt.plot(np.arange(0,numsteps)*dt/600,f(V[3,:],kV[3]),'orange',linewidth = 1)
plt.title('respiratory oscillator')
plt.show()

### WEEK 4:¶

Make a ring attactor network and get it to move

In [10]:
display(HTML("<table><tr><td><img src='x587.png'></td><td><img src=''></td></tr></table>"))
In [19]:
import numpy as np
import matplotlib.pyplot as plt
N = 100
h0 = 2
h1 = 1
T = 2
tao = 1
step = 0.01
numsteps = 1000
w1 = 1
w0 = 0.9

def rect(x):
x[x <= 0] = 0;
a = x;
return(a)

a0 = np.random.rand(N,1)
ptheta = np.ones((N,1))*np.arange(N).reshape((N, 1))
ptheta *= ((2*np.pi)/(N-1))
W = np.zeros((N,N))
for i in range(0,N-1):
for j in range(0,N-1):
W[i,j] = (1/N)*(w0+w1*np.cos(ptheta[i]-ptheta[j]))

a = a0;
ahist = np.zeros((N,numsteps))
for ii in range(0,numsteps):
da = -a + rect(np.dot(W,a)+h0+(h1*np.cos(np.roll(ptheta,ii)))-T)
a += da*step
ahist[:,ii] = a.T
In [30]:
fig = plt.figure(figsize=(6, 6), dpi= 180, facecolor='w', edgecolor='k')
plt.imshow(ahist,cmap = 'RdBu')
Out[30]:
<matplotlib.image.AxesImage at 0x1130eae10>
In [20]:
# First set up the figure, the axis, and the plot element we want to animate
fig = plt.figure(figsize=(5, 3), dpi= 180, facecolor='w', edgecolor='k')
ax = plt.axes(xlim=(0, 100),ylim=(0, 1.5))

line, = ax.plot([], [], lw=2, color = 'k',marker = 'o')
ax.grid()

# initialization function: plot the background of each frame
def init():
line.set_data([], [])
line2.set_data([], [])
line3.set_data([], [])
return line, line2, line3

# animation function.  This is called sequentially
def animate(i):
x = np.arange(0.,100.)
y = ahist[:,i]
line.set_data(x, y)
#line2.set_data(v[0,i+1], v[1,i+1])
#line3.set_data([0,h[0,i+1]], [0,h[1,i+1]])
return line,

# call the animator.  blit=True means only re-draw the parts that have changed.
anim = animation.FuncAnimation(fig, animate, init_func=init,
frames=1000, interval=15, blit=True)

ax.set_title("moving ring attractor bump")

plt.tight_layout()
plt.close()
HTML(anim.to_html5_video())
Out[20]: