Discussion

The model predicts that in PSD-95 knock-out experiments the level of matching between the neurons will decrease. That this is the result of the simulation is no surprise. In the model, the mechanism that matches the orientation after the emergence of selectivity is non-linear, and therefore it is scaled by the selectivity of the neurons. This means that a reduced selectivity decreases the strength of the matching dynamics. With equal amount of noise, an impairment of the selectivity will therefore represent a higher susceptibility to the fluctuations, decreasing the level of orientation matching.

There can be different interpretations if the knock-out experiments are performed and the results

agree with the prediction of the model. Many interpretations would include a fundamental role of PSD-95 in the biological mechanisms responsible for the binocular orientation matching. This model shows that if PSD-95 affects the selectivity of the neurons, the degree of matching between the eyes will also be affected, even if there is no direct causal link between the two.

Chapter 4

Functional implications of the interspersed layout

In the previous part the necessary conditions for the development of interspersed layouts of orien-tation preference were derived and successfully tested in a biologically inspired model of oriented receptive field formation. Analytically it was concluded that, with strong local inhibition, both the unselective state and ordered patterns are unstable solutions, making a disordered layout the only possible outcome in development. In this part of the thesis the resulting pattern will be analyzed numerically. The following questions will be answered:

• How are disordered patterns dynamically formed?

• What are the dynamical implications for a neuron embedded in an disordered network?

• Are all disordered layouts equivalent? Is there a spatial structure in the disorder?

• What is the functional benefit of having disordered layouts in the visual cortex?

This part of the thesis is split in two sections. In the first section the dynamical implications of the interspersed layout are explained. It is shown that the developmental dynamics are similar to a glass system, where frustration results in a power law decrease of changes in orientation and energy with time (aging). It is also shown that the high number of disordered solutions has a strong dynamical implication in the susceptibility of neurons to noise, where even small fluctuations will have a strong effect on the stability of the preferred orientation of neurons. In the second section the spatial char-acteristics of the interspersed pattern are examined in detail. It is shown that a negative correlation in the tuning of the neurons develops, where neurons will tend to have an orientation preference as different as possible from its nearest neighbors. This correlation as a determinant characteristic of interspersed patterns is challenged by the two facts: i) it scales inversely with the number of neu-rons inside the interaction range, and ii) random patterns with the same negative correlation show a higher amount of change in development as the solutions achieved by the dynamics considered here.

Two alternative higher order characterizations are used that are sensitive to different patterns with the same correlation. The fist one is the discrepancy of the layout, measuring the homogeneity of the disorder for any possible pattern interval. The second one has a similar background but a direct

biological interpretation: the Swindale coverage of the pattern. It measures the homogeneity of the representation by the pattern of any given oriented visual input. Both the discrepancy and coverage are improved in the dynamically generated interspersed patterns. The later result leads in chap-ter5of the thesis to propose an evolutionary transition mechanism between maps and interspersed layouts.

4.1 Dynamical characterization

From the analysis of the dynamical equations one can determine the stability of stationary solutions by calculating the growth rate of a perturbation over them. But how the system develops towards that solution is a non-equilibrium physics problem that can’t be solved with the mathematical tools used in the previous part of the thesis. Instead, simulations of a developing pattern are implemented, keeping track of the history of each neuron. Having this information one can calculate how much in average each neuron changes its orientation over time and how the energy of the pattern is decreased.

Many of the simulations in this section were performed in a one dimensional system of 1024 neurons without orientation selective interactions following the numerical methods described in section2.3.3.

Selectivity in the interactions was not included since it was shown in the previous part not to be crucial in the determination of the interspersed patterns. Also recent experimental findings [124]

show that the selective interaction between excitatory neurons in mouse visual cortex emerges only later in development after visual experience. The results presented here are not strongly modified when orientation selective interactions are used. Using this reduction, the dynamics in equation2.1.4 simplifies to

The energy of a layoutzin the system can be calculated using ^∂E_∂z =−^∂z_∂t, obtaining E= −^r

∑

A one dimensional system was chosen because of the possibility to detect symmetrically equivalent patterns. Two patterns are equivalent if a combination of a rotation Rβ, a translationTy and phase shifte^iϕmakes both patterns equal. If how much a pattern changes in time is measured, a develop-ment towards an equivalent pattern should be regarded as no change. The optimal phase shiftϕthat minimizes the angular difference between the neurons in a patternz and a patternyis determined by minimizing the following expression:

To minimize the angular difference the last two terms have to be maximized. For this we define B=

∑

i

ziy_i =|^B|^eiφ

4.1 Dynamical characterization 69

Figure 4.1.1: The development of the interspersed layout is analogous to a glass system where aging in the dynamics takes place. Black curves: Simulation of the system following the dynamics in equation2.1.4. Gray curves: Monte Carlo approach where the selectivity is fixed and for one neuron at the time the orientation is set that minimizes the energy. a) Mean absolute orientation difference of the system compared to the pattern at 1τ.

For Monte Carlo simulations the iterations are discrete and the comparison is done with the orientations of the initial condition. b) Energy decrease in time / Monte Carlo iterations. c) Development of the mean selectivity of the neurons. The one dimensional system is composed of 1024 with interaction parametersS = ¹₄,W = ³₄ andg=0.

and insert in the last two terms, obtaining

∑

i

ziy_ie^−iϕ+

∑

i

ziyie^iϕ =B e^−iϕ+B e^iϕ =|^B|^{e−i(ϕ−φ)}+e^i(ϕ−φ)

=2|^B|^cos(ϕ−^φ) The two terms are maximized with the phase shift

ϕ=φ=arg(

∑

i

ziy_i)

In a one dimensional system rotations do not exist and all possible translations with their correspond-ing optimal orientation shift can be tested simultaneously uscorrespond-ing toeplitz matrices, makcorrespond-ing it possible to find the least difference between two patterns very efficiently. Once again, running the simulations in a two dimensional system doesn’t modify the results.