Notes

A Model of the Ganglion Cell Receptive Field

Assumptions Implicit in the last 3 Slides:

• Receptive fields are difference of gaussians.
• Responses are a weighted average of the stimulus intensity, where the map of the weights is the receptive field.

Are these assumptions reasonable? The second assumption is true if and only if the cell is a linear system.

Linear systems L(x) obey:

• Homogeneity: L(a*x) = a L(x)
• Superposition (Additivity): L(x+y) = L(x) + L(y)

Homogeneity It means that if you have an input to the system x and you measure the response of the system to the input L(x), then this response satisfies: L(a*x) = a*L(x). The picture to the right presents this property by showing a plot of the retinal surface against the neural response, which evidences how the excitation elicited by an input scales linearly.

Superposition If you measure the response of the system to an input x: L(x) and consecutively to an input y: L(y). Then, if the inputs are presented together, the response of the system satisfies: L(x + y) = L(x) + L(y). The same discussion performed for the figure above applies here, which indeed shows the behaviour that the retinal surface should follow to satisfy the superposition property.

Linearity is often checked by using sinusoidal stimuli, because for a linear system:

• The responses to sinusoids are sinusoids.
• The dependence of response on stimulus frequency can be predicted from the shape of the receptive field.

So, if any of these two are false, the system is not linear.

Responses of a Linear System to Sinusoids Sinusoidal stimuli are a sequence of black/white bars that follows a sinusoidal distribution presented synchronously with different intensities.

A Sinusoid in 2-D: A Sinusoidal Grating

Predictions of the Linear Model with a "Difference of Gaussians" Receptive Field If you have a linear system, you can predict the particular response of each ganglion cell receptive field (center/surround/different of center and surround) to a specific spatial frequency, with the knowledge of its difference of gaussians.

Spatial frequency: within one degree of visual angle, how many sinusoidal cycles do we have?

Fitting the Model to the Data

In the figure we have measures of ganglion cells receptive fields responses to changes in spatial frequency, which were made on an on-center cell. They presented sinusoidal stimuli of different spatial frequency, from which we see that by increasing the spatial frequency we obtain a frequency which is optimal for the on-center and then drops again. The fits are good: the responses to sinusoids are predictable by a linear model with a "difference of gaussians" receptive field. Let's try another test of linearity. If it succeeds as well, we'll be happy with the model.