Subsections

• 9.1.1 Homogeneous solutions
• 9.1.2 Stability of homogeneous states
  • 9.1.2.1 Example: Purely excitatory coupling
  • 9.1.2.2 Example: `Mexican-hat' coupling with zero mean
  
  
• 9.1.3 `Blobs' of activity: inhomogeneous states
  • 9.1.3.1 Example: An application to orientation selectivity in V1

9.1 Stationary patterns of neuronal activity

We start with a generic example of pattern formation in a neural network with `Mexican-hat' shaped lateral coupling, i.e., local excitation and long-range inhibition. In order to keep the notation as simple as possible, we will use the field equation derived in Chapter 6; cf. Eq. (6.129). As we have seen in Fig. 6.8, this equation neglects rapid transients and oscillations that could be captured by the full integral equations. On the other hand, in the limit of high noise and short refractoriness the approximation of population dynamics by differential equations is good; cf. Chapter 7. Exact solutions in the low-noise limit will be discussed in Section 9.3.

Consider a single sheet of densely packed neurons. We assume that all neurons are alike and that the connectivity is homogeneous and isotropic, i.e., that the coupling strength of two neurons is a function of their distance only. We loosely define a quantity u(x, t) as the average membrane potential of the group of neurons located at position x at time t. We have seen in Chapter 6 that in the stationary state the `activity' of a population of neurons is strictly given by the single-neuron gain function A₀(x) = g[u₀(x)]; cf. Fig. 9.1. If we assume that changes of the input potential are slow enough so that the population always remains in a state of incoherent firing, then we can set

A(x, t) = g[u(x, t)] ,  (9.1)

even for time-dependent situations. According to Eq. (9.1), the activity A(x, t) of the population around location x is a function of the potential at that location.

The synaptic input current to a given neuron depends on the level of activity of its presynaptic neurons and on the strength of the synaptic couplings. We assume that the amplitude of the input current is simply the presynaptic activity scaled by the average coupling strength of these neurons. The total input current I^syn(x, t) to a neuron at position x is therefore

$$\int \left(\vphantom{\left \vert x-y\right \vert }\right. \left\vert\vphantom{x-y}\right. \left.\vphantom{x-y}\right\vert \left.\vphantom{\left \vert x-y\right \vert }\right)$$ (9.2)

Here, w is the average coupling strength of two neurons as a function of their distance. We consider a connectivity pattern, that is excitatory for proximal neurons and predominantly inhibitory for distal neurons. Figure B shows the typical `Mexican-hat shape' of the corresponding coupling function. Eq. (9.2) assumes that synaptic interaction is instantaneous. In a more detailed model we could include the axonal transmission delay and synaptic time constants. In that case, A(y, t) on the right-hand side of Eq. (9.2) should be replaced by $\int$$\alpha$(s) A(y, t - s) ds where $\alpha$(s) is the temporal interaction kernel.

Figure 9.1: A. Generic form of the sigmoidal gain function g of graded response neurons that describes the relation between the potential u and the `activity' of the neural population. B. Typical `Mexican hat'-shaped function that is used here as an ansatz for the coupling w of two neurons as a function of their distance x.

In order to complete the definition of the model, we need to specify a relation between the input current and the resulting membrane potential. In order to keep things simple we treat each neuron as a leaky integrator. The input potential is thus given by a differential equation of the form

$$\tau {\frac{{\partial u}}{{\partial t}}}$$ (9.3)

with $\tau$ being the time constant of the integrator and I^ext an additional external input. If we put things together we obtain the field equation

$$\tau {\frac{{\partial u(x,t)}}{{\partial t}}} \int \left(\vphantom{\left \vert x-y\right \vert }\right. \left\vert\vphantom{x-y}\right. \left.\vphantom{x-y}\right\vert \left.\vphantom{\left \vert x-y\right \vert }\right)$$ (9.4)

cf. Amari (1977b); Feldman and Cowan (1975); Wilson and Cowan (1973); Kishimoto and Amari (1979). This is a nonlinear integro-differential equation for the average membrane potential u(x, t).

9.1.1 Homogeneous solutions

Although we have kept the above model as simple as possible, the field equation (9.4) is complicated enough to prevent comprehensive analytical treatment. We therefore start our investigation by looking for a special type of solution, i.e., a solution that is uniform over space, but not necessarily constant over time. We call this the homogenous solution and write u(x, t) $\equiv$ u(t). We expect that a homogenous solution exists if the external input is homogeneous as well, i.e., if I^ext(x, t) $\equiv$ I^ext(t).

Substitution of the ansatz u(x, t) $\equiv$ u(t) into Eq. (9.4) yields

$$\tau {\frac{{{\text{d}}u(t)}}{{{\text{d}}t}}} \bar{{w}}$$ (9.5)

with $\bar{{w}}$ = $\int$dy  w$\left(\vphantom{\left \vert y \right \vert }\right.$$\left\vert\vphantom{ y }\right.$y$\left.\vphantom{ y }\right\vert$$\left.\vphantom{\left \vert y \right \vert }\right)$. This is a nonlinear ordinary differential equation for the average membrane potential u(t). We note that the equation for the homogeneous solution is identical to that of a single population without spatial structure; cf. Eq. (6.87) in Chapter 6.3.

The fixed points of the above equation with I^ext = 0 are of particular interest because they correspond to a resting state of the network. More generally, we search for stationary solutions for a given constant external input I^ext(x, t) $\equiv$ I^ext. The fixed points of Eq. (9.5) are solutions of

$${\frac{{u-I^{\text{ext}}}}{{\bar{w}}}}$$ (9.6)

which is represented graphically in Fig. 9.2. Depending on the strength of the external input three qualitatively different situations can be observed. For low external stimulation there is a single fixed point at a very low level of neuronal activity. This corresponds to a quiescent state where the activity of the whole network has ceased. Large stimulation results in a fixed point at an almost saturated level of activity which corresponds to a state where all neurons are firing at their maximum rate. Intermediate values of external stimulation, however, may result in a situation with more than one fixed point. Depending on the shape of the output function and the mean synaptic coupling strength $\bar{{w}}$ three fixed points may appear. Two of them correspond to the quiescent and the highly activated state, respectively, which are separated by the third fixed point at an intermediate level of activity.

Figure 9.2: Graphical representation of the fixed-point equation (9.6). The solid line corresponds to the neuronal gain function g(u) and the dashed lines to (u - I^ext)/$\bar{{w}}$ for different amounts of external stimulation I^ext. Depending on the amount of I^ext there is either a stable fixed point at low activity (leftmost black dot), a stable fixed point at high activity (rightmost black dot), or a bistable situation with stable fixed points (black dots on center line) separated by an unstable fixed point at intermediate level of activity (small circle).

Any potential physical relevance of fixed points clearly depends on their stability. Stability under the dynamics defined by the ordinary differential equation Eq. (9.5) is readily checked using standard analysis. Stability requires that at the intersection

$$\bar{{w}}^{{-1}}_{}$$ (9.7)

Thus all fixed points corresponding to quiescent or highly activated states are stable whereas the middle fixed point in case of multiple solutions is unstable; cf. Fig. 9.2. This, however, is only half of the truth because Eq. (9.5) only describes homogeneous solutions. Therefore, it may well be that the solutions are stable with respect to Eq. (9.5), but unstable with respect to inhomogeneous perturbations, i.e., to perturbations that do not have the same amplitude everywhere in the net.

9.1.2 Stability of homogeneous states

In the following we will perform a linear stability analysis of the homogeneous solutions found in the previous section. To this end we study the field equation (9.4) and consider small perturbations about the homogeneous solution. A linearization of the field equation will lead to a linear differential equation for the amplitude of the perturbation. The homogeneous solution is said to be stable if the amplitude of every small perturbation is decreasing whatever its shape.

Suppose u(x, t) $\equiv$ u₀ is a homogeneous solution of Eq. (9.4), i.e.,

$$\int \left(\vphantom{\left \vert x-y\right \vert }\right. \left\vert\vphantom{x-y}\right. \left.\vphantom{x-y}\right\vert \left.\vphantom{\left \vert x-y\right \vert }\right)$$ (9.8)

Consider a small perturbation $\delta$u(x, t) with initial amplitude $\left\vert\vphantom{ \delta u(x,0) }\right.$$\delta$u(x, 0)$\left.\vphantom{ \delta u(x,0) }\right\vert$ $\ll$ 1. We substitute u(x, t) = u₀ + $\delta$u(x, t) in Eq. (9.4) and linearize with respect to $\delta$u,

$$\begin{multline} \tau \, \frac{\partial}{\partial t} \delta u(x,t) = -u_0 - \... ...a u(y,t)] + I^{\text{ext}}(x,t) +{\mathcal{O}}(\delta u^2) \,. \end{multline}$$

Here, a prime denotes the derivative with respect to the argument. Zero-order terms cancel each other because of Eq. (9.8). If we collect all terms linear in $\delta$u we find

$$\tau {\frac{{\partial}}{{\partial t}}} \delta \delta \int \delta$$ (9.9)

We make two important observations. First, Eq. (9.10) is linear in the perturbations $\delta$u - simply because we have neglected terms of order ($\delta$u)ⁿ with n$\ge$2. Second, the coupling between neurons at locations x and y is mediated by the coupling kernel w$\left(\vphantom{\left \vert x-y\right \vert }\right.$$\left\vert\vphantom{ x-y }\right.$x - y$\left.\vphantom{ x-y }\right\vert$$\left.\vphantom{\left \vert x-y\right \vert }\right)$ that depends only on the distance | x - y|. If we apply a Fourier transform over the spatial coordinates, the convolution integral turns into a simple multiplication. It suffices therefore to discuss a single (spatial) Fourier component of $\delta$u(x, t). Any specific initial form of $\delta$u(x, 0) can be created from its Fourier components by virtue of the superposition principle. We can therefore proceed without loss of generality by considering a single Fourier component, viz., $\delta$u(x, t) = c(t) e^i k x. If we substitute this ansatz in Eq. (9.10) we obtain

$$\tau \left[\vphantom{ 1 - g'(u_0) \, \int \!\! {\text{d}}y \; w(\vert x-y\vert) \, {\text{e}}^{i\,k \, (y-x)} }\right. \int \left.\vphantom{ 1 - g'(u_0) \, \int \!\! {\text{d}}y \; w(\vert x-y\vert) \, {\text{e}}^{i\,k \, (y-x)} }\right]$$

$$\left[\vphantom{ 1 - g'(u_0) \, \int \!\! {\text{d}}z \; w(\vert z\vert) \, {\text{e}}^{i\,k \, z} }\right. \int \left.\vphantom{ 1 - g'(u_0) \, \int \!\! {\text{d}}z \; w(\vert z\vert) \, {\text{e}}^{i\,k \, z} }\right]$$ (9.10)

which is a linear differential equation for the amplitude c of a perturbation with wave number k. This equation is solved by

$$\kappa$$ (9.11)

with

$$\kappa \int$$ (9.12)

Stability of the solution u₀ with respect to a perturbation with wave number k depends on the sign of the real part of $\kappa$(k). Note that - quite intuitively - only two quantities enter this expression, namely the slope of the activation function evaluated at u₀ and the Fourier transform of the coupling function w evaluated at k. If the real part of the Fourier transform of w stays below 1/g'(u₀), then u₀ is stable. Note that Eqs. (9.12) and (9.13) are valid for an arbitrary coupling function w(| x - y|). In the following two examples we illustrate the typical behavior for two specific choices of the lateral coupling.

9.1.2.1 Example: Purely excitatory coupling

We now apply Eq. (9.13) to a network with purely excitatory couplings. For the sake of simplicity we take a one-dimensional sheet of neurons and assume that the coupling function is bell-shaped, i.e.,

$${\frac{{\bar{w}}}{{\sqrt{2 \pi \, \sigma^2}}}} \sigma^{2}$$ (9.13)

with the mean strength $\int$dx  w(x) = $\bar{{w}}$ and characteristic length scale $\sigma$. The Fourier transform of w is

$$\int \bar{{w}} \sigma^{2}$$ (9.14)

with maximum $\bar{{w}}$ at k = 0. According to Eq. (9.13) a homogeneous solution u₀ is stable if 1 - $\bar{{w}}$ g'(u₀)  > 0. This is precisely the result obtained by the simple stability analysis based on the homogeneous field equation; cf. Eq. (9.7). This result indicates that no particularly interesting phenomena will arise in networks with purely excitatory coupling.

9.1.2.2 Example: `Mexican-hat' coupling with zero mean

A more interesting example is provided by a combination of excitatory and inhibitory coupling described by the difference of two bell-shaped functions with different width. For the sake of simplicity we will again consider a one-dimensional sheet of neurons. For the lateral coupling we take

$${\frac{{\sigma_2 \, {\text{e}}^{-x^2/(2 \sigma_1^2)} - \sigma_1 \, {\text{e}}^{-x^2/(2 \sigma_2^2)}}}{{\sigma_2-\sigma_1}}}$$ (9.15)

with $\sigma_{1}^{}$ < $\sigma_{2}^{}$. The normalization of the coupling function has been chosen so that w(0) = 1 and $\int$dx  w(x) = $\bar{{w}}$ = 0; cf Fig. 9.3A.

Figure 9.3: A. Synaptic coupling function with zero mean as in Eq. (9.16) with $\sigma_{1}^{}$ = 1 and $\sigma_{2}^{}$ = 10. B. Fourier transform of the coupling function shown in A; cf. Eq. (9.18). C. Gain function g(u) = {1 + exp[$\beta$(x - $\theta$)]}^-1 with $\beta$ = 5 and $\theta$ = 1. The dashing indicates that part of the graph where the slope exceeds the critical slope s ^* . D. Derivative of the gain function shown in C (solid line) and critical slope s ^* (dashed line).

As a first step we search for a homogeneous solution. If we substitute u(x, t) = u(t) in Eq. (9.4) we find

$$\tau {\frac{{{\text{d}}u(t)}}{{{\text{d}}t}}}$$ (9.16)

The term containing the integral drops out because of $\bar{{w}}$ = 0. This differential equation has a single stable fixed point at u₀ = I^ext. This situation corresponds to the graphical solution of Fig. 9.2 with the dashed lines replaced by vertical lines (`infinite slope').

We still have to check the stability of the homogenous solution u(x, t) = u₀ with respect to inhomogeneous perturbations. In the present case, the Fourier transform of w,

$$\int {\frac{{\sqrt{2 \pi} \, \sigma_1 \, \sigma_2}}{{\sigma_2-\sigma_1}}} \left(\vphantom{ {\text{e}}^{-k^2\,\sigma_1^2/2} - {\text{e}}^{-k^2\,\sigma_2^2/2} }\right. \sigma_{1}^{2} \sigma_{2}^{2} \left.\vphantom{ {\text{e}}^{-k^2\,\sigma_1^2/2} - {\text{e}}^{-k^2\,\sigma_2^2/2} }\right)$$ (9.17)

vanishes at k = 0 and has its maxima at

$$\left[\vphantom{ \frac{2 \, \ln (\sigma_2^2/\sigma_1^2)}{\sigma_2^2-\sigma_1^2} }\right. {\frac{{2 \, \ln (\sigma_2^2/\sigma_1^2)}}{{\sigma_2^2-\sigma_1^2}}} \left.\vphantom{ \frac{2 \, \ln (\sigma_2^2/\sigma_1^2)}{\sigma_2^2-\sigma_1^2} }\right]^{{1/2}}_{}$$ (9.18)

At the maximum, the amplitude of the Fourier transform has a value of

$$\hat{{w}}_{m}^{} \max_{k}^{} \int {\frac{{\sqrt{2 \pi} \, \sigma_1 \, \sigma_2}}{{\sigma_2-\sigma_1}}} \left[\vphantom{ \left( \frac{\sigma_1^2}{\sigma_2^2} \right )^{\... ...a_1^2}{\sigma_2^2} \right )^{\frac{\sigma_2^2}{\sigma_2^2-\sigma_1^2}} }\right. \left(\vphantom{ \frac{\sigma_1^2}{\sigma_2^2} }\right. {\frac{{\sigma_1^2}}{{\sigma_2^2}}} \left.\vphantom{ \frac{\sigma_1^2}{\sigma_2^2} }\right)^{{\frac{\sigma_1^2}{\sigma_2^2-\sigma_1^2}}}_{} \left(\vphantom{ \frac{\sigma_1^2}{\sigma_2^2} }\right. {\frac{{\sigma_1^2}}{{\sigma_2^2}}} \left.\vphantom{ \frac{\sigma_1^2}{\sigma_2^2} }\right)^{{\frac{\sigma_2^2}{\sigma_2^2-\sigma_1^2}}}_{} \left.\vphantom{ \left( \frac{\sigma_1^2}{\sigma_2^2} \right )^{\... ...a_1^2}{\sigma_2^2} \right )^{\frac{\sigma_2^2}{\sigma_2^2-\sigma_1^2}} }\right]$$ (9.19)

cf. Fig. 9.3B. We use this result in Eqs. (9.12) and (9.13) and conclude that stable homogeneous solutions can only be found for those parts of the graph of the output function f (u) where the slope s = g'(u) does not exceed the critical value s ^* = 1/$\hat{{w}}_{m}^{}$,

$${\frac{{\sigma_2-\sigma_1}}{{\sqrt{2 \pi} \, \sigma_1 \, \sigma_2}}} \left[\vphantom{ \left( \frac{\sigma_1^2}{\sigma_2^2} \right )^{\... ...a_1^2}{\sigma_2^2} \right )^{\frac{\sigma_2^2}{\sigma_2^2-\sigma_1^2}} }\right. \left(\vphantom{ \frac{\sigma_1^2}{\sigma_2^2} }\right. {\frac{{\sigma_1^2}}{{\sigma_2^2}}} \left.\vphantom{ \frac{\sigma_1^2}{\sigma_2^2} }\right)^{{\frac{\sigma_1^2}{\sigma_2^2-\sigma_1^2}}}_{} \left(\vphantom{ \frac{\sigma_1^2}{\sigma_2^2} }\right. {\frac{{\sigma_1^2}}{{\sigma_2^2}}} \left.\vphantom{ \frac{\sigma_1^2}{\sigma_2^2} }\right)^{{\frac{\sigma_2^2}{\sigma_2^2-\sigma_1^2}}}_{} \left.\vphantom{ \left( \frac{\sigma_1^2}{\sigma_2^2} \right )^{\... ...gma_2^2} \right )^{\frac{\sigma_2^2}{\sigma_2^2-\sigma_1^2}} }\right]^{{-1}}_{}$$ (9.20)

Figure Fig. 9.3 shows that depending on the choice of coupling w and gain functions g a certain interval for the external input exists without a corresponding stable homogeneous solution. In this parameter domain a phenomenon called pattern formation can be observed: Small fluctuations around the homogeneous state grow exponentially until a characteristic pattern of regions with low and high activity has developed; cf. Fig. 9.4.

Figure 9.4: Spontaneous pattern formation in a one-dimensional sheet of neurons with `mexican-hat' type of interaction and homogeneous external stimulation. The parameters for the coupling function and the output function are the same as in Fig. . The graphs show the evolution in time of the spatial distribution of the average membrane potential u(x, t). A. For I^ext = 0.4 the homogeneous low-activity state is stable, but it looses stability at I^ext = 0.6 (B). Here, small initial fluctuations in the membrane potential grow exponentially and result in a global pattern of regions with high and low activity. C. Similar situation as in B, but with I^ext = 1.4. D. Finally, at I^ext = 1.6 the homogeneous high-activity mode is stable.

9.1.3 `Blobs' of activity: inhomogeneous states

From a computational point of view bistable systems are of particular interest because they can be used as `memory units'. For example a homogeneous population of neurons with all-to-all connections can exhibit a bistable behavior where either all neurons are quiescent or all neurons are firing at their maximum rate. By switching between the inactive and the active state, the neuronal population would be able to represent, store, or retrieve one bit of information. The exciting question that arises now is whether a neuronal net with distance-dependent coupling w(| x - y|) can store more than just a single bit of information, but spatial patterns of activity. Sensory input, e.g., visual stimulation, could switch part of the network to its excited state whereas the unstimulated part would remain in its resting state. Due to bistability this pattern of activity could be preserved even if the stimulation is turned off again and thus provide a neuronal correlate of working memory.

Let us suppose we prepare the network in a state where neurons in one spatial domain are active and all remaining neurons are quiescent. Will the network stay in that configuration? In other words, we are looking for an `interesting' stationary solution u(x) of the field equation (9.4). The borderline where quiescent and active domains of the network meet is obviously most critical to the function of the network as a memory device. To start with the simplest case with a single borderline, we consider a one-dimensional spatial pattern where the activity changes at x = 0 from the low-activity to the high-activity state. This pattern could be the result of inhomogeneous stimulation in the past, but since we are interested in a memory state we now assume that the external input is simply constant, i.e., I^ext(x, t) = I^ext. Substitution of the ansatz u(x, t) = u(x) into the field equation yields

$$\int$$ (9.21)

This is a nonlinear integral equation for the unknown function u(x).

We can find a particular solution of Eq. (9.22) if we replace the output function by a simple step function, e.g.,

$$\begin{cases} 0\,, & u<\vartheta \\ 1\,, & u \ge \vartheta\, . \end{cases}$$ (9.22)

In this case g[u(x)] is either zero or one and we can exploit translation invariance to define g[u(x)] = 1 for x > 0 and g[u(x)] = 0 for x < 0 without loss of generality. The right-hand side of Eq. () does now no longer depend on u and we find

$$\int_{{-\infty}}^{x}$$ (9.23)

and in particular

$${\frac{{1}}{{2}}} \bar{{w}}$$ (9.24)

with $\bar{{w}}$ = $\int$dy  w(| y|). We have calculated this solution under the assumption that g[u(x)] = 1 for x > 0 and g[u(x)] = 0 for x < 0. This assumption imposes a self-consistency condition on the solution, namely that the membrane potential reaches the threshold $\vartheta$ at x = 0. A solution in form of a stationary border between quiescent and active neurons can therefore only be found, if

$$\vartheta {\frac{{1}}{{2}}} \bar{{w}}$$ (9.25)

If the external stimulation is either smaller or greater than this critical value, then the border will propagate to the right or to the left.

Following the same line of reasoning, we can also look for a localized `blob' of activity. Assuming that g[u(x)] = 1 for x $\in$ [x₁, x₂] and g[u(x)] = 0 outside this interval leads to a self-consistency condition of the form

$$\vartheta \int_{0}^{\Delta}$$ (9.26)

with $\Delta$ = x₂ - x₁. The mathematical arguments are qualitatively the same, if we replace the step function by a more realistic smooth gain function.

Figure 9.5 shows that solutions in the form of sharply localized excitations exist for a broad range of external stimulations. A simple argument also shows that the width $\Delta$ of the blob is stable if w($\Delta$) < 0 (Amari, 1977b). In this case blobs of activity can be induced without the need of fine tuning the parameters in order to fulfill the self-consistency condition, because the width of the blob will adjust itself until stationarity is reached and Eq. (9.27) holds; cf. Fig. 9.5A.

Figure 9.5: Localized `blobs' of activity. A. A small initial perturbation develops into a stable blob of activity. B. Stationary profile of a localized excitation for various amounts of external stimulation ( I^ext = 0, 0.5,..., 0.3 in order of increasing width). Note that for strong stimuli neurons in the center of the activity blob are less active than those close to the edge of the blob.

9.1.3.1 Example: An application to orientation selectivity in V1

Stable localized blobs of activity may not only be related to memory states but also to the processing of sensory information. A nice example is the model of Ben-Yishai et al. (1995) [see also Hansel and Sompolinsky (1998)] that aims at a description of orientation selectivity in the visual cortex.

It is found experimentally that cells from the primary visual cortex (V1) respond preferentially to lines or bars that have a certain orientation within the visual field. There are neurons that `prefer' vertical bars, others respond maximally to bars with a different orientation (Hubel, 1995). Up to now it is still a matter of debate where this orientation selectivity does come from. It may be the result of the wiring of the input to the visual cortex, i.e., the wiring of the projections from the LGN to V1, or it may result from intra-cortical connections, i.e., from the wiring of the neurons within V1, or both. Here we will investigate the extent to which intra-cortical projections can contribute to orientation selectivity.

We consider a network of neurons forming a so-called hypercolumn. These are neurons with receptive fields which correspond to roughly the same zone in the visual field but with different preferred orientations. The orientation of a bar at a given position within the visual field can thus be coded faithfully by the population activity of the neurons from the corresponding hypercolumn.

Instead of using spatial coordinates to identify a neuron in the cortex, we label the neurons in this section by their preferred orientation $\theta$ which may vary from - $\pi$/2 to + $\pi$/2. In doing so we assume that the preferred orientation is indeed a good ``name tag'' for each neuron so that the synaptic coupling strength can be given in terms of the preferred orientations of pre- and postsynaptic neuron. Following the formalism developed in the previous sections, we assume that the synaptic coupling strength w of neurons with preferred orientation $\theta$ and $\theta{^\prime}$ is a symmetric function of the difference $\theta$ - $\theta{^\prime}$, i.e., w = w(|$\theta$ - $\theta{^\prime}$|). Since we are dealing with angles from [- $\pi$/2, + $\pi$/2] it is natural to assume that all functions are $\pi$-periodic so that we can use Fourier series to characterize them. Non-trivial results are obtained even if we retain only the first two Fourier components of the coupling function,

$$\theta \theta{^\prime} \theta \theta{^\prime}$$ (9.27)

Similarly to the intra-cortical projections we take the (stationary) external input from the LGN as a function of the difference of the preferred orientation $\theta$ and the orientation of the stimulus $\theta_{0}^{}$,

$$\theta \theta \theta_{0}^{}$$ (9.28)

Here, c₀ is the mean of the input and c₂ describes the modulation of the input that arises from anisotropies in the projections from the LGN to V1.

In analogy to Eq. (9.4) the field equation for the present setup has thus the form

$$\tau {\frac{{\partial u(\theta,t)}}{{\partial t}}} \theta \int_{{-\pi/2}}^{{+\pi/2}} {\frac{{{\text{d}}\theta'}}{{\pi}}} \theta \theta{^\prime} \theta{^\prime} \theta$$ (9.29)

We are interested in the distribution of the neuronal activity within the hypercolumn as it arises from a stationary external stimulus with orientation $\theta_{0}^{}$. This will allow us to study the role of intra-cortical projections in sharpening orientation selectivity.

In order to obtain conclusive results we have to specify the form of the gain function g. A particularly simple case is the step-linear function,

$$\equiv \begin{cases} u\,, & u \ge 0 \\ 0\,, & u < 0 \end{cases}$$ (9.30)

The idea behind this ansatz is that neuronal firing usually increases monotonously once the input exceeds a certain threshold. Within certain boundaries this increase in the firing rate is approximatively linear. If we assume that the average membrane potential u stays within these boundaries and that, in addition, u($\theta$, t) is always above threshold, then we can replace the gain function g in Eq. (9.30) by the identity function. We are thus left with the following linear equation for the stationary distribution of the average membrane potential,

$$\theta \int_{{-\pi/2}}^{{+\pi/2}} {\frac{{{\text{d}}\theta'}}{{\pi}}} \theta \theta{^\prime} \theta{^\prime} \theta$$ (9.31)

This equation is solved by

$$\theta \theta \theta_{0}^{}$$ (9.32)

with

$${\frac{{c_0}}{{1-w_0}}} {\frac{{2\,c_2}}{{2-w_2}}}$$ (9.33)

As a result of the intra-cortical projections, the modulation u₂ of the response of the neurons from the hypercolumn is thus amplified by a factor 2/(2 - w₂) as compared to the modulation of the input c₂.

Figure 9.6: Activity profiles (solid line) that result from stationary external stimulation (dashed line) in a model of orientation selectivity. A. Weak modulation (c₀ = 0.8, c₂ = 0.2) of the external input results in a broad activity profile; cf. eqstat theta. B. Strong modulation (c₀ = 0.6, c₂ = 0.4) produces a narrow profile; cf. Eq. (9.35). Other parameters are $\omega_{0}^{}$ = 0, $\omega_{2}^{}$ = 1, $\theta_{0}^{}$ = 0.

In deriving Eq. (9.32) we have assumed that u stays always above threshold so that we have an additional condition, viz., u₀ - | u₂| > 0, in order to obtain a self-consistent solution. This condition may be violated depending on the stimulus. In that case the above solution is no longer valid and we have to take the nonlinearity of the gain function into account, i.e., we have to replace Eq. (9.32) by

$$\theta \int_{{\theta_0-\theta_c}}^{{\theta_0+\theta_c}} {\frac{{{\text{d}}\theta'}}{{\pi}}} \theta \theta{^\prime} \theta{^\prime} \theta$$ (9.34)

Here, $\theta_{0}^{}$±$\theta_{c}^{}$ are the cutoff angles that define the interval where u($\theta$) is positive. If we use the ansatz (9.33) in the above equation, we obtain together with u($\theta_{0}^{}$±$\theta_{c}^{}$) = 0 a set of equations that can be solved for u₀, u₂, and $\theta_{c}^{}$. Figure 9.6 shows two examples of the resulting activity profiles g[u($\theta$)] for different modulation depths of the input.

Throughout this section we have described neuronal populations in terms of an averaged membrane potential and the corresponding firing rate. At least for stationary input and a high level of noise this is indeed a good approximation of the dynamics of spiking neurons. Figure 9.7 shows two examples of a simulation based on SRM₀ neurons with escape noise and a network architecture that is equivalent to what we have used above. The stationary activity profiles shown in Fig. 9.7 are qualitatively identical to those of Fig. 9.6, small deviations in the shape are due to a slightly different model [see Spiridon and Gerstner (2001) for more details]. For low levels of noise, however, the description in terms of a firing rate is no longer valid, because the state of asynchronous firing becomes unstable (cf. Section 8.1) and neurons tend to synchronize. The arising temporal structure in the firing times leads to a destabilization of the stationary spatial structure (Laing and Chow, 2001). In the low-noise limit localized ``blobs'' of activity are replaced by traveling waves of spike activity, as we will see in Section 9.3.

Figure 9.7: Activity profiles in a model of orientation selectivity obtained by simulations based on SRM₀ neurons (dots) compared to the theoretical prediction (solid line). A. No modulation of the recurrent projections [ $\omega_{2}^{}$ = 0; cf. Eq. (9.28)] leads only to a weak modulation of the neuronal response. B. Excitatory coupling between iso-oriented cells ( $\omega_{2}^{}$ = 10) produces a sharp profile. [Taken from Spiridon and Gerstner (2001)].