Cortex is a large, but thin sheet of neurons. Field models, in their spatial interpretation, describe the population activity of neurons as a function of the location on the cortical sheet.
Field models are, however, also used in a more general setting. In sensory cortices, neuronal activity encodes continuous variables, such as position of an object, orientation of edges, direction of movement etc. Field models, in their more abstract interpretation, represent the distribution of activity along the axes representing one or several of these variables.
In field models, interactions between populations of neurons depend on the distance of neurons in the physical, or abstract, space. Classical field models assume a Mexican-hat interaction pattern where local excitation is combined with long-range inhibition. Field models with Mexican-hat interaction have two important regimes of parameter settings. In the input-driven regime, spatial activity patterns can only arise if the input has a non-trivial spatial structure. In the bump-attractor regime, however, localized blobs of activity emerge even in the absence of input.
Mexican-hat interaction combines excitation and inhibition from the same presynaptic population and must therefore be considered as an effective, mathematical coupling scheme between neurons. It is, however, possible to construct similar field models with separate populations of excitatory neurons with long-range, and inhibitory neurons with short-range interactions. These models bring the main results of field models a step closer to biological reality.
Bump formation . Consider a one-dimensional discrete recurrent network with population units $1\leq i\leq N$ and update rule
$A_{i}(t+1)=F(\sum_{k}w_{ik}x_{k}(t)+\sum_{j}B_{ij}A_{j}(t))$ | (18.35) |
where $w_{ik}$ is the coupling strength to the input $x_{k}$ and $B_{ij}$ are the recurrent weights. Each neuron receives local excitation from its $d$ neighbors on both sides: $B_{ij}=1$ for $|i-j|\leq d$ and inhibition from all others $B_{ij}=-\beta\leq-1$ for $|i-j|>d$ . The gain function $F(h)$ is the Heaviside step function, i.e. $F(h)=1$ for $h>0$ and $F(h)=0$ for $h\leq 0$ .
(i) Imagine that one single unit is stimulated and therefore becomes active. This neuron will excite its neighbors. Show that in the steady state of the network the number $N$ of active neurons is larger than $2d$ .
Hint: Consider the balance of excitation and inhibition at the border of the blob.
(ii) How does the value of $\beta$ influence the number of active neurons? What happens in the limit of $\beta\to 1$ ?
(iii) Assume that $d=5$ , $N=1000$ and the input to neuron $i=17$ is 1. Compute the first three time steps of the network dynamics.
Stability of homogeneous solution with excitatory coupling
(i) Consider the purely excitatory coupling
$w(x)=\frac{\bar{w}}{\sqrt{2\pi\,\sigma^{2}}}\,{\text{e}}^{-x^{2}/(2\sigma^{2})% }\,,$ | (18.36) |
with the mean strength $\int{\text{d}}x\;w(x)=\bar{w}$ and Fourier transform
$\int\!\!{\text{d}}x\,w(x)\,{\text{e}}^{i\,k\,x}=\bar{w}\,{\text{e}}^{-k^{2}\,% \sigma^{2}/2}\,,$ | (18.37) |
Under what conditions is the homogeneous solution stable (assume $F^{\prime}(h_{0})>0$ )?
(ii) Consider a general coupling function $w(x)$ such that this function can be written as an autocorrelation
$w(x)=\bar{w}\int_{-\infty}^{\infty}f(x^{\prime}-x)f(x^{\prime})dx^{\prime}\,,$ | (18.38) |
for some real function $f(x)$ . Under what conditions is the homogeneous solution stable? (Hint: The convolution theorem).
Phase plane analysis of inhibition-stabilized network . An excitatory population is coupled to an inhibitory population, controlled by the activity equations
$\displaystyle\tau_{E}{dA_{E}\over dt}$ | $\displaystyle=$ | $\displaystyle-A_{E}+F(w_{EE}A_{E}-w_{EI}A_{I}+I_{E})$ | |||
$\displaystyle\tau_{I}{dA_{I}\over dt}$ | $\displaystyle=$ | $\displaystyle-A_{I}+F(w_{IE}A_{E}-w_{II}A_{I}+I_{I}-\vartheta)$ | (18.39) |
Assume that $F(h)=0$ for $h<0$ ; $F(h)=h$ for $0\leq h\leq 1$ and $F(h)=1$ for $h>1$ .
(i) Draw the nullclines in the phase plane spanned by the variables $A_{E}$ ( $x$ -axis) and $A_{I}$ ( $y$ -axis) in the absence of input $I_{E}=I_{I}=0$ and $\vartheta=0.5$ . Assume that $w_{EE}=w_{EI}=2$ , $w_{IE}=1$ and $w_{II}=0$ .
(ii) Assume that the inhibitory population receives positive input $I_{I}=0.2$ . Redraw the nullclines. Does the population activity of excitatory or inhibitory populations in crease or decrease? Does this correspond to your intuition? Can you interpret the result?
Surround inhibition . We study the model of the previous exercise in the linear region ( $F(h)=h)$ . Two instantiations $i=1,2$ of the model are coupled via an additional connection from $A_{E,1}$ to $A_{i,2}$ and from $A_{E,2}$ to $A_{i,1}$ with lateral connections $w_{\rm lat}>0$ .
(i) Assume that the first excitatory population receives an input $I_{E,1}=0.3$ . Calculate the stationary population activity of all four populations.
(ii) Assume that both excitatory populations receive an input $I_{E,1}=0.3=I_{E,2}$ . Calculate the stationary population activity of $A_{E,1}$ and $A_{I,1}$ . Do the population activities increase or decrease compared to the case considered in (i)?
(iii) Can you interpret your result in the context of surround suppression?
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