Understanding visual map formation through vortex dynamics of spin Hamiltonian models Myoung Won Cho BK21 Frontier Physics Research Division department of Physics & Astronomy Seoul National University

Functional maps in the brain Primary visual cortex (V1) Primary auditory cortex (A1) Motor cortex Somatosensory cortex Ocular dominance map Orientation preference map

Primary visual cortex (V1)

Orientation preference (OP) map in V1 Smoothly varying OPs in an array Optical imaging technique reveals the detailed whole feature of visual cortex OP map of tree shrew (Bosking et al 1997) Pinwheel (singular point) Linear zone (periodic patterns)

Ocular dominance (OD) map in V1 Ocular (left or right eye) dominance column The structure of the primary visual cortex

Red : external stimuli B(x) Blue : neighbor interactions through lateral connections J(x,y) J(x,y) = J(|x-y|) : Mexican hat type where S i = (S i x,S i y )=(cos 2  i, sin 2  i ) for preferred angle  i or S i = (S i z ) for ocular dominance S i z Spike-like Hamiltonian model of OP and OD maps k : degree of negative interactions

- There are so many models which can lead successful visual map formations. ex) Miller’s correlation-based model, Spin-like Hamiltonian model, Kohonen’s SOFM, Elastic net model, … - It may be needed to determine the proper model by comparing with more detailed experimental data. (Erwin et al. 1995, Swindale 1996) However, we cannot determined the proper one by comparing with simulation results because the statistical characteristics of visual maps do not originate from the unique neural mechanisms. (Cho & Kim, 2005)

Universal theory of cortical map formations Cho & Kim PRL (2005)

What determines the typical characteristics of cortical maps?

The typical characteristics of cortical maps are determined by the topologic properties of lattice and phase space rather than detailed neural interaction or learning rules. They share the statistical properties with other physical systems having the same topology: OD map => Ising spin model (O(1) symmetry) OP map => XY spin model (O(2) symmetry) OP+OD map => Heisenberg spin model (O(3) symmetry)

Pinwheels in OP map ≡ Vortices in XY spin system (Singularities in 2D lattice systems having O(2) symmetry) (Wolf & Geisel, Nature, 1998)

Autocorrelation function of preferred orientation in Macaque (Obermayer 1993) Emergence of periodic patterns by large inhibitory connections between long-distance neurons

Perpendicularity at cortical boundary OD map of macaque monkey OP map of tree shrew Electrostatic fields near conductor or

Correlation between OP and OD maps Experimental results show OP and OD maps are correlated - contour lines intersect perpendicularly - island structures in OD map locate near the singularity of OP map OD OP

Joint model of OP and OD maps

Competitive relationship between OP and OD components => Development of strong OD components near pinwheels =0.62 =0.65 MagnetismVisual cortex Orientation singularity In-plane vortex Pinwheel in OP Scalar peak near singularity Out-of-plane vortex Island pattern in OD Anisotropy  ~ J OD / J OP <= No periodic pattern because of weak negative interactions

Different OD map formations depending on the anisotropy Ising model region XY model region Pure Heisenberg model region Out-of-plane vortex unstable OD segregation strength Cho & Kim, PRL (2005)

Three typical visual map types observed in different animals - island patterns in OD maps - intermediate OD segregation - Stripe patterns in OD maps - Strong OD segregation - Weak OD segregation (no OD map) - Strong orientation selectivity

Pinwheel stability depending on the anisotropy (Wolf & Geisel, Nature, 1998)

- Collective behaviors in a cortex system can be clarified by the general principles in statistical mechanics in spite of unique neural mechanisms and complex cortical structures. - Symmetry property is one of the keywords to understand collective behaviors in a neural system like other physical systems.