3.3 Inference and Decision Making
4.1.2 Mean Field Approximation
The approach can be extended to networks with a spatial connectivity structure (Amari, 1977). Mathematically speaking, the underlying idea is a socalled continuum limit, i.e. the assumption that the number of neurons is so high, that individual neurons can safely be replaced by a neuron density and the interaction is mediated by a field. So instead of speaking of the membrane potential viof the
neuron i at position xi, we think of the membrane potential as a function of space
v(x). This makes it possible to capture distance-dependent connection densities betweeen neurons in a kernel w(kx − yk), where x and y are the positions of neurons. Applying the same principles as in the previous subsection, we can formulate the neural field equation:
τ d
dtv(x) = −v(x) +
Z ∞
−∞
w(|x − y|)f v(y) dy + b (4.9)
Note that this equation can also be formulated for multiple populations as we have done in equation (4.2). For the sake of simplicity, however, we constrain ourselves to the single population case here.
4.1. MEAN RATE APPROACHES 53 Spatial Patterns of Activity in Inhibitory Networks
Analogously to the previous section, we can compute the stationary solution by setting the left hand side of equation (4.9) to zero. This yields the equation
v(x) =
Z ∞
−∞
w(|x − y|)f v(y) dy + b. (4.10)
This integral equation has one trivial solution v(x) = const. = v0. The equation
then simplifies massively and the value v0 can be computed in a similar way as
before:
v0=
Z ∞
−∞
w(|x − y|)f (v0) dy + b = f (v0) ¯w + b (4.11)
Here, ¯w is used as a shorthand for R w(|x − y|) dy. As in the previous section, ¯w is a measure for the absolute connection strength, justifying the repetition in notation.
Notably, the form of this solution is independent of the shape of the connection kernel. Its stability, however, crucially depends on the shape of the connection kernel w(|x − y|). Consider two different shapes: Firstly, a gaussian bell curve around the center
wGauss(∆x) = ¯ w √ 2πσ2exp −∆x 2 2σ2 . (4.12)
Here, the connection density decays as the distance ∆x from the neuron increases and the parameter σ is a measure for how wide the neuron’s connections reach in space (see figure 4.2). Also note that in keeping with the previous definition of ¯w, the connection kernel is normalized such that its integral equals ¯w.
Secondly, consider a symmetric gamma-distribution shaped connection kernel
wGamma(∆x) = ¯ w|∆x|n−1exp−|∆x| θ 2θnΓ(n) , n > 1. (4.13)
With this connection profile, the connection density starts at 0 then increases to a maximum at a distance of ∆x = (n − 1)θ, after which it decays to zeros (see figure 4.2)). While the gaussian connection kernel has maximal connectivity close to the center, the gamma kernel has maximal connectivity at a distance determined by n and θ. Accordingly, the gaussian is an example of socalled on-center-inhibition, while the gamma-kernel is off-center inhibition (seeRinzel,
1998).
54 CHAPTER 4. NEURAL DYNAMICS
Figure 4.2: Spatial patterns of activity. a) Connection kernels: Gaussian (red) and Gamma-kernel (blue) b) Fourier transforms of the connection kernels. c Numerical solution for equation (4.15) for Gaussian (red) and Gamma-kernel (blue).
perturbation around v(x) = v0= cont.:
v(x) = v0+ ν(x, t). (4.14)
In this equation is a very small (compared to v0) perturbation parameter and
ν(x) is an arbitrary, but bounded function of position x and time t.
The dynamics can then be computed by inserting equation (4.14) into equa- tion (4.9). This yields
τ d dt v0+ ν(x) = − v0+ ν(x) + Z ∞ −∞ w(|x − y|)f v0+ ν(y) dy + b τ d dt v0+ ν(x) = −v0− ν(x) + Z w(|x − y|)f (v0) + df dv v=v 0 ν(y)dy + b τ d dtν(x) = −ν(x) + Z w(|x − y|)df dv v=v 0 ν(y) dy. (4.15) In the first step, it was exploited that the perturbation is assumed small and the transfer function f (·) can be linearized around v0. Subsequently, in the second
step, the stationarity condition for v0, τdtdv0= −v0+ ¯wf (v0) + b, was subtracted
and afterwards was divided out.
The resultant integral equation in (4.15) can be treated much better in the Fourier domain. For this purpose, a Fourier transform with respect to space x is applied to both sides of the equation, yielding
τd dtF T [ν](k) = −F T [ν](k) + F T [w](k) df dv v=v 0 F T [ν](k) τ d dteν(k) = −ν(k) +e w(k)e df dv v=v 0 e ν(k). (4.16)
The second equation in (4.16) introduces the shorthand notationeν for F T [ν]. k is the spatial frequency. Importantly, in the Fourier domain, the integral term, a convolution of w(x) and ν, becomes a product, which allows for an analytical
4.1. MEAN RATE APPROACHES 55 solution.
We can now investigate the stability for each frequency component k sepa- rately. Based on the same reasoning as before, the condition for stability ofνe for any frequency component k is
d dνe −eν(k) +w(k)e df dv v=v 0e ν(k)< 0 −1 +w(k)e df dv v=v 0 < 0. (4.17)
Note the similarity of this equation with (4.7). In the case of an inhibitory network, w(|x − y|) is negative. Moreover, assume the transfer function is monotonically increasing and therefore dfdv v=v
0 is positive, irrespective of v.
Hence, ifw(k) is negative for all k, the condition in equation (4.17) is alwayse fulfilled. It follows, that, in the case of an inhibitory network, a necessary condition for constant-rate solution to be unstable is thatw(k) has positive parts.e More generally, stability in a network with distance specific connection density depends crucially on the shape of the Fourier transform of the connection profile w(|x − y|).
The Fourier transformsw for both the gaussian we gauss(|x−y|) and the gamma
wgamma(|x − y|) connection kernel can be computed analytically. They are given
by e wgauss(k) = ¯w exp −1 2(σk) 2 e wgamma(k) = ¯w< 1 (1 + ikθ)n (4.18)
Figure reffig:meanfield shows these fourier transform (without ¯w)s. The gaussian transforms into another gaussian, the width of which is inversely related to the width of the original gaussian. So the transform of a gaussian kernel is positive for all values of k. The Fourier transform of the gamma kernel, on the other hand, has negative parts at a non-zero frequqency. This means for an inhibitory connection kernel ( ¯w < 0),wegamma(k) takes positive values for some frequency
components k. If, in addition, the slope of the transfer function dfdv v=v
0
is high enough for condition (4.17) to not hold anymore, the spatially homogeneous solution v(x) = v0= const. becomes unstable. The minima ofwegamma(k) are the
frequency components which increase the strongest after the perturbation and hence determine the spatial periodicity of the emerging pattern. This minima can be computed as
kc= arg min
k wegamma(k) =
tann+1π
56 CHAPTER 4. NEURAL DYNAMICS To summarize, the analysis shows that stability crucially depends on three factors: Firstly, the slope of the transfer function at the stationary point dfdv
v=v
0
needs to be sufficiently high for instabilities to emerge. Secondly, the absolute strength ¯w of the connections contributes in the same way, i.e., high enough ¯w is a condition for instability. These two factors are the same as in the mean rate approach. In addition, for the mean field approach with distance dependent connectivity, the shape of the connection kernel is pivotal.
In a purely inhibitory network, this means specifically that only a connection kernel w(|x − y|), of which the Fourier transformw(k) has negative parts at non-e zero frequencies, like the gamma kernel, can lead to the spatially homogeneous solution v(x) = v0 = const. becoming unstable. The network then converges
towards a solution with spatial periodicity with frequency kc in (4.19). In other
words, this means that stable bumps of high activity form at equal distances given by 2π
kc. In two or three dimensions, these bumps are arranged in hexagonal
(2D) or tetrahedral (3D) pattern. Spreizer (2016) analyzed the emergence of these bumps in numerical simulations and corroborated the outlined analysis.