Localized Solutions in a Simple Neural Field Model J. Michael Herrmann, Hecke Schrobsdorff, Theo Geisel MPI f¨ urStr¨omungsforschungG¨ottingen and Universit¨ at G¨ ottingen Dept. of Nonlinear Dynamics, Bunsenstr. 10, 37073 G¨ ottingen michael|hecke|geisel@chaos.gwdg.de Abstract We investigate analytically properties like stability and existence of solutions of the two dimensional neural field equation as proposed by Amari (1977) in [1] as a model of macroscopic activation dynamics in neural tissue. While the one dimensional case has been treated comprehensively, for the two dimensional case only the existence of circular solutions was shown, and stability was as well only considered for radially symmetric perturbations. We introduce a simplified neural field model, defined by a piecewise constant interaction kernel. First we show that in this setting the existence of noncircular solutions can be negated except for small per- turbations of a circle. Secondly in a special case, the dynamics of the boundary of an activated area is considered, which leads to a proof, that in this special case, solutions are necessarily circular. Neural Fields Neural field equation cf. [1]: τ ∂u(x,t) ∂t = −u(x,t)+  R[u] w (|x − x ′ |)dx ′ + h (1) R[u]= {x |u(x) > 0} is the excited region. Equilibrium configurations obey: u(x,t)=  R[u] w (|x − x ′ |)dx ′ + h (2) 1D: kernel w Resulting fixpoints: ⋆ Periodic patterns ⋆ Localized solutions 2D: Here the situation is almost the same as in the 1D-case, cf. [4]. Simulations result in circular symmetry → → → → t =0 t = ∞ New effects in two di- mensions, cf. [5]: ring solutions → → t =0 t = ∞ Simplified Model Piecewise constant interaction function w (g + ,g - > 0) w (r )=      g + r ≤ R E −g − R E <r ≤ R I 0 otherwise kernel: ⋆ Reduction to elementary geometrical problem ⋆ Boundary points: g + A + = g − A + activated region K of a solution of (2): We consider a parameter region, which allows for localized solutions. It is R E < 2R . Restrictions to Perturbations Assumptions: ⋆ No neutral areas: R I > 2R ⋆ S is connected ⋆ transversal intersection of K and B Then the minimum return angle converges to ϕ min λ→0 = R E R . discontinuous at λ =0 Maximum perturbations of circle solu- tions depending on the ratio of radii R E R : 1 1 2 3 1 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Maximum Amplitude Ratio of Radii 0 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Angle between a and b Ratio of Radii Boundary Dynamics Further assumption: ⋆ R = R E , thus ratio of weights: g + g − = 2π +3 √ 3 4π − 3 √ 3 ≈ 1.558 ⋆ third order of ‖c 1 − c 2 ‖ and thus the curvature of ∂S are neglected. The situation leads to the dependence of ‖a 2 − M ‖ and ‖b 2 − M ‖ on the geometrical parameters: a x 2 a y 1 + b x 1 b y 2 + c x 2 (a y 2 + b y 2 )+ c y 1 (a x 1 + b x 1 )+2z x (b y 1 + c y 2 )+2z y (b x 2 - c x 1 ) +r c ( arcsin ( (a 1 - c 1 ) · (z - c 1 ) r -2 c ) + arcsin ( (b 2 - c 2 ) · (z - c 2 ) r -2 c )) = a x 1 a y 2 + b x 2 b y 1 + c y 2 (a x 2 + b x 2 )+ c x 1 (a y 1 + b y 1 )+2z x (b y 2 + c y 1 )+2z y (b x 1 - c x 2 ) +r c ( arcsin ( (a 2 - c 2 ) · (z - c 2 ) r -2 c ) + arcsin ( (b 1 - c 1 ) · (z - c 1 ) r -2 c )) By a coordinate transform, this leads to a differential equation for the radial distance of ∂S to M . For r c R E =1 it simplifies to: ∂y (ϕ) ∂t = −2  y (ϕ) · (4 − y (ϕ)) solution -4 -2 2 4 x -1.5 -1 -0.5 0.5 1 1.5 y Results from boundary dynamics: ⋆ The model is consistent with circular solutions. ⋆ For almost all ratios of radii the boundary points move discontinu- ously. ⋆ For r c R E =1 and r c R E = √ 5 a continuous change is observed, but then the resulting boundary curves lead to disconnected activated areas. Conclusion ⋆ Although computational complexity may show limiting in further elaboration of the approach, in our model we have justified the con- jecture [4, 5] that localized solutions have to be circularly symmetric, by setting hard limits to the possibly effective perturbations. ⋆ For another special case, we could show, that indeed, no noncircular connected solutions exist. ⋆ In this context, we have shown, how circularity can be reduced to local behavior of the boundary. References [1] S. Amari: Dynamics of pattern formation in lateral-inhibition type neural fields. Biological Cybernetics 27 (1977) 77-87. [2] K. Kishimoto, S. Amari: Existence and stability of local excitations in homogeneous neural fields. Mathematical Biology 7 (1979) 303-318. [3] C. R. Laing, W. C. Troy: PDE Methods for Non-Local Models. SIAM Journal of Applied Dynamical Systems 2:3 (2003) 487-516. [4] J. G. Taylor: Neural ’bubble’ dynamics in two dimensions: foundations. Biological Cybernetics 80 (1999) 393-409. [5] H. Werner, T. Richter: Circular stationary solutions in two-dimensional neural fields. Biological Cybernetics 85 (2001) 211-217.