Extended difference-of-Gaussians model incorporating cortical feedback for relay cells in the lateral geniculate nucleus of cat

Abstract

A striking feature of the organization of the early visual pathway is the significant feedback from primary visual cortex to cells in the dorsal lateral geniculate nucleus (LGN). Despite numerous experimental and modeling studies, the functional role for this feedback remains elusive. We present a new firing-rate-based model for LGN relay cells in cat, explicitly accounting for thalamocortical loop effects. The established DOG model, here assumed to account for the spatial aspects of the feedforward processing of visual stimuli, is extended to incorporate the influence of thalamocortical loops including a full set of orientation-selective cortical cell populations. Assuming a phase-reversed push-pull arrangement of ON and OFF cortical feedback as seen experimentally, this extended DOG (eDOG) model exhibits linear firing properties despite non-linear firing characteristics of the corticothalamic cells. The spatiotemporal receptive field of the eDOG model has a simple algebraic structure in Fourier space, while the real-space receptive field, as well as responses to visual stimuli, are found by evaluation of an integral. As an example application we use the eDOG model to study effects of cortical feedback on responses to flashing circular spots and patch-grating stimuli and find that the eDOG model can qualitatively account for experimental findings.

Appendices

Appendix 1: eDOG model for small cortical feedback strengths

For $|C|<1$, the denominator term of the integral expression for f _eDOG(r) in Eq. 35 can be replaced by a series expansion, i.e., $1/(1-x)= \sum_{m=0}^\infty x^m = 1 + x + x^2 + \ldots$. With $x = C e^{-k^2 c^2/4}$ we find by insertion into Eq. 35:

$$ \begin{aligned} f_{{\rm eDOG}}(r) &= \frac{1}{(2 \pi)^2} \iint{{\bf k}} e^{i {\bf k} {\bf r}} \frac{A_1e^{-k^2 a_1^2/4}- A_2 e^{-k^2 a_2^2/4}} {1 - Ce^{-k^2 c^2/4}} \hbox{d}^2{\bf k}\\ &= \frac{1}{(2 \pi)^2} \iint{{\bf k}} e^{i {\bf k} {\bf r}} \left( A_1e^{-k^2 a_1^2/4}- A_2 e^{-k^2 a_2^2/4} \right) \times \sum_{m=0}^\infty C^{m}e^{-m k^2 c^2/4}\hbox{d}^2{\bf k}\\ &= \sum_{m=0}^\infty \frac{C^m}{(2 \pi)^2} \iint{{\bf k}} e^{i {\bf k} {\bf r}} \left( A_1 e^{-k^2(a_1^2+mc^2)/4} - A_2 e^{-k^2(a_2^2+mc^2)/4} \right) \hbox{d}^2{\bf k}\\ &= \sum_{m=0}^\infty C^{m} \left( A_1 \frac{e^{-r^2/\left(a_{1}^2+mc^2 \right)}}{\pi (a_{1}^2+mc^2)} -A_2 \frac{e^{-r^2/\left(a_{2}^2+mc^2 \right)}}{\pi (a_{2}^2+mc^2)} \right). \end{aligned} $$

(40)

The final expression in Eq. 40 has a relatively simple form: it consists of an infinite sum of DOGs. For small values of C, the higher terms vanish rapidly, so that only a limited number of terms need to be retained. The terms in the sum have increasing spatial extent ($\sqrt{a_1^2 + mc^2}, \sqrt{a_2^2 + mc^2}$) and alternating sign when C is negative.

The terms in the series have an intuitive interpretation as illustrated by the first three terms:

$$ \begin{aligned} f_{{\rm eDOG}}(r) &= \left( \frac{A_1}{\pi a_1^2} e^{-r^2 /a_1^2} - \frac{A_2}{\pi a_2^2} e^{-r^2 /a_2^2} \right) \quad\quad (m=0) \\ &\quad + C \left( \frac{A_1}{\pi(a_1^2+ c^2)} e^{-r^2 /(a_1^2 + c^2)} - \frac{A_2}{\pi(a_2^2+ c^2)} e^{-r^2 /(a_2^2 + c^2)} \right) \quad\quad (m=1) \\ &\quad+ C^2 \left( \frac{A_1}{\pi(a_1^2+ 2 c^2)} e^{-r^2 /(a_1^2 + 2 c^2)} - \frac{A_2}{\pi(a_2^2+ 2 c^2)} e^{-r^2/(a_2^2+ 2 c^2)} \right) \quad\quad (m=2) \\ &\quad+\cdots\\ \end{aligned} $$

The first term corresponds to the “direct” feedforward DOG. The second term corresponds to the correction to this “direct” term due to one round of the feedforward DOG in the cortical feedback loop. The spatial extension of the Gaussians is larger since they have been ”spread out” by the spatial spread of the corticogeniculate loop. Correspondingly, the third term corresponds to a “correction of the correction” after the direct term has gone through the thalamocortical loop twice. This term is spatially even more spread out after two rounds through the loop.

For $|C|<1$, the series in Eq. 40 can be shown to be uniformly convergent assuring the validity of the mathematical derivation. If the feedback is positive with C = 1, the integrand in Eq. 35 has a singular point corresponding to a resonance in the thalamocortical loop (Einevoll and Plesser 2002).

Appendix 2: Numerical methods

Figures 2, 3 were created using Mathematica 7 (Wolfram Research), while Figs. 4, 6 were created using Matlab 7.10.0 (MathWorks) on a Lenovo T410i computer running Windows XP.