The effect of ICMS is described in [35] (and references therein). Directly at the ICMS site the orientation preference remains constant. In its close vicinity, the preferred orientation is shifted towards the orientation at the ICMS site. Thus, locally ICMS has a enforcing effect on the orientation preference. The cortical region subject to the direct effect of ICMS is approximately 100µm which is an order of magnitude smaller than the spacing of orientation columns.
The effect of ICMS was examined in the phenomenological model (5.1) for the activity-driven dynamics of orientation maps introduced in Chapter 2. As outlined in Chapter 2, this model exhibits stable solutions which resemble orientation maps in the visual cortex. Since the aim is an understanding of the reorganization in the adult animal, the effect of ICMS on such stable solution is sought. To mimic the situation of an ICMS experiment, we modeled the application of ICMS by enforcing locally the preferred orientation. The full dynamics including the action of ICMS at cortical locationxS reads
∂tz(x,t) = Lz(x,t) +N[z(x,t)] +Θ(t)Θ(tS−t)S(x) (7.1)
whereΘis the Heaviside function, tS the duration of ICMS (starting att =0). The first two terms on the r.h.s. denote the original model (5.1). A stable solution of the free dynamics will be denoted byz∞(x)in the following.
In the model, ICMS is mimicked by an external force term. Consistent with the phenomenology of the experiment, ICMS shall enforce the preferred orientation at the ICMS site within a small cortical region. We thus modeled ICMS by choosing
S(x) =δr z∞(xS)µ(x−xS) (7.2) whereδis the ICMS strength andµthe stimulation function describing the spatial form of the direct ICMS effect. In the following, we use
µ(x) = 1
a Gaussian with widthσS. The direct effect of the force term Eq. (7.2) is to strengthen the selectivity|z(xS)|around the stimulation sitexS and to widen the surrounding iso-orientation domain. Thus, for a given orientation map z∞ and ICMS site xS, ICMS is controlled by three parameters: The strengthδand durationtS of applica-tion, and the spatial widthσS. The width is constraint byσS Λsince the exten-sion of the direct ICMS effect is expected to be small on the scale of the columnar patternΛ. The durationtS shall be small on the scalet∗ on which a state z∞ close to the attractor is reached.
The implementation of ICMS in the form (7.2) can be interpreted as a simplifica-tion of a more accurate descripsimplifica-tion. Including modulasimplifica-tions of the map z∞ in the vicinity of the ICMS site and accounting for the temporal evolution of the stimula-tion the ICMS term may be written as
S1(x,t) =δrz(x−xS,t)µ(x−xS,t) (7.4) with z(x,t = 0) = z∞(x). Assuming that properties of the tissue (location of neurons, the surrounding glia cells, the concentration of ions, etc.) remain stable, the stimulation function can be taken as constant in time, µ(x,t) ' µ(x). If, in addition, the stimulation has a small range, the direction of stimulation can be ap-proximated throughz(x) ' z(xS) by the direction at the center. At pinwheel cen-ters where the orientation preference is not continuous the selectivity approaches zero, such that the ICMS term becomes very small. Since the ICMS term enforces the selectivity aroundxS, the phase of z(xS) will remain constant expecially if the complete system is translation and rotation invariant. Simplifying, one can use z(xS,t) ' z∞(xS) thereby avoiding the positive feedback-loop in Eq. (7.4) for which there is no experimental evidence and which is undesirable from a tech-nical perspective. Thus, apart from pinwheel centers where the modeling of ICMS is unclear for several reasons, both implementations Eq. (7.2) and Eq. (7.4) are
expected to yield similar results. As will become apparent below, the basic mecha-nism underlying the effect of ICMS on the map is common for both forms.
7.3 Methods
7.3.1 Numerical integration
The semi-implicit numerical integration scheme used to integrate Eq. (7.1) without the ICMS term and its implementation are described in Section 7.2. The ICMS part (7.2) was treated explicitely. Writing
N˜0 =N˜ +S˜ (7.5)
it was implemented in the Fourier domain as an additional term of the nonlinear part ˜N of Eq. (5.10). Formally, ˜N0 was treated as a correcting force term to the dynamics governed by the linear partL.
ICMS was applied to a near final state integrated until t = 106 (in time scales oft = 1/r) using strengths ranging from δ = 10−2.5 toδ = 10−1 with durations between tS = 102 and tS = 103.5. After the termination of ICMS, the solution was advanced over t = 104 to obtain a new stable solutions z. To calculate the induced modification∆(x), the ICMS site was varied on a 50×50 grid in an area of 6.6Λ×6.6Λwithin a larger map of size 17Λ×17Λ.
We used amplitude equations to study systematically the dependence on the ICMS parameter. Amplitude equations were integrated by an Adams-Gear scheme based on backward differentiation formula methods. The method is accurate in 5th order. The Jacobian matrix was approximated by divided differences. In parameter regimes with stable solutions of n active modes the integration was performed for 2n amplitudes by approximating the interaction coefficients by gii = 1, gi j = g, gii− = 2, fi j6=i = g/2 and fii− = 1. Before and after ICMS the solution was advanced tot=1011. Further parameters wereg =0.98, andr =0.1. Consistency test were carried out with arbitrary number of amplitudes (between 32 and 128) and the complete coefficient matrix.
7.3.2 Pinwheel analysis
Pinwheels were identified and tracked over time as described in Section 6.2.1.
States were monitored at times ti separated by exponentially increasing time in-tervals ∆ti = ti−ti−1 between t = 0 and t = tS during the period of ICMS and betweent =tS and t =tS+104during relaxation after ICMS. A total of N =100 maps was sampled during each period.
Figure 7.1: Transient changes induced by intracortical microstimulation (ICMS). Orientation preference mapϑ(x)(OPM), orientation selectivity|z(x)|(OSI), and 2D power spectrum|z˜(k)|2 (origin of k-space at the center of annulus) before (Pre) and at different stages after the applica-tion of ICMS (Post-Final state). Shown is a subregions of a larger map (Γ=17) integrated on a 128×128grid withg=0.98,σ =1.7,δ=10−2.9,tS =102.7, advanced tot =106before and t = 104 after ICMS. For each state the local cross-correlation (Eq. (7.7)) with the initial map is shown in the lower row. Bright (dark) regions exhibit larger (smaller) correlations. Note that the initial and the final maps are very similar (average cross correlations: Pre/Post,r = 0.71; Pre/Final,r=0.95) and exhibit identical power-spectra.