Configuring Neural Ensembles

When creating a neural ensemble, a variety of parameters can be adjusted. Some of these parameters are set to reflect the physiological properties of the neurons being modelled, while others can be set to improve the accuracy of the transformations computed by the neurons.

Membrane time constant and refractory period

The two parameters for Leaky-Integrate-and-Fire neurons are the membrane time constant (tau_rc) and the refractory period (tau_ref). These parameters are set when creating the ensemble, and default to 0.02 seconds for the membrane time constant and 0.002 seconds for the refractory period:

net.make('D',100,2,tau_rc=0.02,tau_ref=0.002)

Empirical data on the membrane time constants for different types of neurons in different parts of the brain can be found at http://ctn.uwaterloo.ca/~cnrglab/?q=node/547.

Maximum firing rate

You can also specify the maximum firing rate for the neurons. It should be noted that it will always be possible to force these neurons to fire faster than this specified rate. Indeed, the actual maximum firing rate will always be 1/tau_ref, since if enough current is forced into the simulated neuron, it will fire as fast as its refractory period will allow. However, what we can specify with this parameter is the normal operating range for the neurons. More technically, this is the maximum firing rate assuming that the neurons are representing a value within the ensemble’s radius.

In most cases, we specify this by giving a range of maximum firing rates, and each neuron will have a maximum chosen uniformly from within this range. This gives a somewhat biologically realistic amount of diversity in the tuning curves. The following line makes neurons with maximums between 200Hz and 400Hz:

net.make('E',100,2,max_rate=(200,400))

Alternatively, we can specify a particular set of maximum firing rates, and each neuron will take on a value from the provided list. If there are more neurons than elements in the list, the provided values will be re-used:

net.make('F',100,2,max_rate=[200,250,300,350,400])

Note

The type of brackets used is important!! Python has two types of brackets for this sort of situation: round brackets () and square brackets []. Round brackets create a tuple, which we use for indicating a range of values to randomly choose within, and square brackets create a list, which we use for specifying a list of particular value to use.

Intercept

The intercept is the point on the tuning curve graph where the neuron starts firing. For example, for a one-dimensional ensemble, a neuron with a preferred direction vector of [1] and an intercept of 0.3 will only fire when representing values above 0.3. If the preferred direction vector is [-1], then it will only fire for values below 0.3. In general, the neuron will only fire if the dot product of x (the value being represented) and the preferred direction vector (see below), divided by the radius, is greater than the intercept. Note that since we divide by the radius, the intercepts will always be normalized to be between -1 and 1.

While this parameter can be used to help match the tuning curves observed in the system being modelled, one important other use is to build neural models that can perfectly represent the value 0. For example, if a 1-dimensional neural ensemble is built with intercepts in the range (0.3,1), then no neurons at all will fire for values between -0.3 and 0.3. This means that any value in that range (i.e. any small value) will be rounded down to exactly 0. This can be useful for optimizing thresholding and other functions where many of the output values are zero.

By default, intercepts are uniformly distributed between -1 and 1. The intercepts can be specified by providing either a range, or a list of values:

net.make('G',100,2,intercept=(-1,1))
net.make('H',100,2,intercept=[-0.8,-0.4,0.4,0.8])

Note

The type of brackets used is important!! Python has two types of brackets for this sort of situation: round brackets () and square brackets []. Round brackets create a tuple, which we use for indicating a range of values to randomly choose within, and square brackets create a list, which we use for specifying a list of particular value to use.

Encoders (a.k.a. preferred direction vectors)

You can specify the encoders (preferred direction vectors) for the neurons. By default, the encoders are chosen uniformly from the unit sphere. Alternatively, you can specify those encoders by providing a list. The encoders given will automatically be normalized to unit length:

net.make('F',100,2,encoders=[[1,0],[-1,0],[0,1],[0,-1]])
net.make('G',100,2,encoders=[[1,1],[1,-1],[-1,1],[-1,-1]])

This allows you to make complex sets of encoders by creating a list with the encoders you want. For example, the following code creates an ensemble with 100 neurons, half of which have encoders chosen from the unit circle, and the other half of which are aligned on the diagonals:

import random
import math

encoders=[]              # create an empty list to store the encoders
for i in range(50):
    theta=random.uniform(-math.pi,pi)  # choose a random direction between -pi and pi
    encoders.append([math.sin(theta),math.cos(theta)])  # add the encoder
for i in range(50):
    encoders.append(random.choice([[1,1],[1,-1],[-1,1],[-1,-1]]))  # add an aligned encoder

net.make('G',100,2,encoders=encoders)     # create the ensemble