Circular symmetry of solutions of the neural field equation
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1 Neural Field Dynamics Circular symmetry of solutions of the neural field equation take 287 Hecke Dynamically Selforganized Max Punk Institute Symposium on Nonlinear Glacier Dynamics 2005 in Zermatt
2 Neural Field Dynamics Outline Introduction Neural field equation The 1D-Case Extensions to 2D Arising problems Simplification of the interaction kernel Proof of circularity Conclusions Outlook
3 Neural Field Dynamics OK, OK! Just Kidding... Not again. And now for something (not completely) different!
4 Localization of Neural Activity a new episode Hecke Schrobsdorff, Vincent David, Joachim Hass, Marc Timme, Michael Herrmann, Theo Geisel Max Planck Institute for Dynamics and Selforganization Symposium on Nonlinear Dynamics 2005 in Zermatt
5 Outline 1 Introduction 2 Numerical Findings 3 Description of the Model 4 Existence of neural breathers 5 Recent results
6 Outline 1 Introduction 2 Numerical Findings 3 Description of the Model 4 Existence of neural breathers 5 Recent results
7 Outline 1 Introduction 2 Numerical Findings 3 Description of the Model 4 Existence of neural breathers 5 Recent results
8 Outline 1 Introduction 2 Numerical Findings 3 Description of the Model 4 Existence of neural breathers 5 Recent results
9 Outline 1 Introduction 2 Numerical Findings 3 Description of the Model 4 Existence of neural breathers 5 Recent results
10 Introduction Introduction Individual Units
11 Introduction Introduction Localized Excitations
12 Numerical Findings Motivation
13 Description of the Model Model of one unit v vth v vreset t Mirollo-Strogatz Neurons (here: leaky integrate and fire) resting potential v threshold potential v th refraction potential v reset
14 Description of the Model Network structure delta coupling spatial organization in a chain equal synaptic weights ǫ only to the two nearest neighbours, all excitatory ǫ ǫ ǫ ǫ synaptic delay τ v i 2 v i 1 v i v i+1 v i+2 network size arbitrary, only localized activations are of interest no learning
15 Description of the Model Network structure delta coupling spatial organization in a chain equal synaptic weights ǫ only to the two nearest neighbours, all excitatory ǫ ǫ ǫ ǫ synaptic delay τ v i 2 v i 1 v i v i+1 v i+2 network size arbitrary, only localized activations are of interest no learning
16 Description of the Model Network structure delta coupling spatial organization in a chain equal synaptic weights ǫ only to the two nearest neighbours, all excitatory ǫ ǫ ǫ ǫ synaptic delay τ v i 2 v i 1 v i v i+1 v i+2 network size arbitrary, only localized activations are of interest no learning
17 Existence of neural breathers Existence ( ) vreset v For τ > ln 1 + th v v th the spike-pattern below exists and is stable for [ 1 ( ǫ v th v + (v th v reset )e γ2τ) [,(1 e γτ )(v th v ) 2 unit τ 2τ 3τ 4τ time
18 Existence of neural breathers Proof v i vth1 v 1 vreset1 vth2 v 2 vreset2 vth3 v 3 vreset3 vth4 v 4 vreset4 vth5 v 5 vreset5 vth6 v 6 vreset6 vth7 v 7 vreset7 vth8 v 8 vreset8 τ 2τ 3τ t
19 Existence of neural breathers Return map Let the crossing of v th be preserved. Return map: ( ) ( ) ( t3,i+1 t3,i max(t3,i, t = F = 5,i ) + 2τ max(t 3,i, t 5,i ) + 2τ t 5,i+1 t 5,i ) This is already linear. No dependence of the concrete potentials because of the fix v reset
20 Existence of neural breathers Return map Let the crossing of v th be preserved. Return map: ( ) ( ) ( t3,i+1 t3,i max(t3,i, t = F = 5,i ) + 2τ max(t 3,i, t 5,i ) + 2τ t 5,i+1 t 5,i ) This is already linear. No dependence of the concrete potentials because of the fix v reset
21 Existence of neural breathers Return map Let the crossing of v th be preserved. Return map: ( ) ( ) ( t3,i+1 t3,i max(t3,i, t = F = 5,i ) + 2τ max(t 3,i, t 5,i ) + 2τ t 5,i+1 t 5,i ) This is already linear. No dependence of the concrete potentials because of the fix v reset
22 Existence of neural breathers Resynchronization v i vth3 v 3 vreset3 vth4 v 4 vreset4 vth5 v 5 vreset5 vth6 v 6 vreset6 τ τ + τ 2τ 2τ + τ 3τ + τ t
23 Recent results Larger patterns unit τ 2τ 3τ 4τ time
24 Recent results larger patterns v intern vthintern v intern vresetintern 3ǫ ǫ τ 2τ 3τ t t
25 Recent results No Resynchronization unit τ 2τ 3τ 4τ time
26 Outlook Outlook I A more general network structure should be considered (Hi Marc!). Inner neurons outer neurons boundary neurons
27 Outlook Outlook II With a good understanding of breezers, a simulational model of associating in PFC is imaginable (Hi Michael!). IT V4 PFC
28 Summary Summary Localized excitation is possible in very simple models. With leaky integrate n fire neurons their existence and stability can be investigated analytically.
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