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## Propagation of activity between two neurons

We have described the basic biophysics of an isolated neuron in the last section. Let us now recall, in a very schematic way, the path of information from one neuron j to another one i. When a spike arrives at a synapse sij;k, depicted by A1 in gure 2.2, it produces a post-synaptic potential PSPij;k as we describe in the previous section. We drop the index of the synapse k between neuron i and j for convenience. This potential is then ltered by the dendritic tree (A1-A2) and the eective potential felt by the soma is a temporally shifted and attenuated version of the original PSPij .

Hence, depending on where the input comes to the dendrite, the potential felt by the soma can be quite dierent. If the somatic membrane potential Vi is above threshold, then a spike is produced at the axon hillock (see A3). This spike is transmitted along the axon, without attenuation, until it reaches another synapse (see A4). The delay it takes for a spike to go from the axon hillock to the synapse is written dji. In the NF model, we suppose that the dendrite is punctual and we neglect the eects of nonlinear spatial ltering that are produced by the dendritic tree (see [Coombes 2003, Venkov 2008] for a NF model of dendrite). We also assume that there is a single synapse to simplify the notations. We can compensate for this assumption by assuming that more spikes come to the synapse between i and j. In particular, we neglect the propagation delays in the dendritic tree although they could be easily introduced, in a heuristic way, in our nal equations (2.12). On the other hand, we have seen above that this synapse introduces an eective delay Dji. Hence, the total delay between the production of the spikes in each neuron is ji Dji + dji.

### Relationship between the two formulations

We have derived two NF models in the last two sections. Based on remark 4, if the synapses are fast, the dominant time constant is the membrane time constant: the voltage-based model is more appropriate. On the other hand, if the membrane time constant is small, the dominant time constant is the synaptic decay time: the activity-based model is more appropriate.

If we suppose that Iext is stationary, and assume that the delays ji are zero and L is scalar L = l Id, then the voltage-based model can be derived from the activity based model the change of variables V(t) = J A(t) + Iext. Also the stationary points of each model are mapped by the change of variables A = S(V).

#### The propagation-delay function

Following the derivation in section 2.1.2, we call (r; r0) the total delay for the processing of the information, from populations located at r0 to populations located at r. What is the space dependent delay function (r; r0) in the cortex? Our analysis is built upon the recent paper [Budd 2010]. It is concluded that if a neuron located at r is connected to another neuron located at r0, the path length of this connection is very close to kr r0k2, the Euclidean distance. In other words, axons are straight lines. This is true if the two neurons are at most 2mm apart but we will assume it is also true for long range connections. Our analysis carries through for more general delay functions. Moreover, we have seen in section 2.1.1 that constant delays have to be introduced in order to take into account the nite integration time of action potentials by synapses ans post-synaptic neurons. Hence, we choose the following delay function:

**Table of contents :**

**I Introduction to the biology/modeling of the visual cortex **

**1 Introduction to the biology of vision **

1.1 The visual pathway in the primates

1.1.1 The eye and the retina

1.1.2 The lateral geniculate nucleus (LGN)

1.1.3 The visual cortex

1.1.4 Selectivity and maps

1.2 Connections

1.2.1 Thalamo-cortical connections

1.2.2 Intra-cortical connections in V1

1.3 A closer look at orientation selectivity

1.3.1 Biological facts

1.3.2 Two classes of models

1.4 Conclusion

**2 The neural elds model **

2.1 A brief account for the ow neuronal activity

2.1.1 Introduction

2.1.2 Propagation of activity between two neurons

2.2 The local models

2.2.1 The voltage-based model

2.2.2 The activity-based model

2.2.3 Relationship between the two formulations

2.2.4 The continuum models

2.2.5 The propagation-delay function

2.3 The Mexican hat model

**II Stationary cortical states **

**3 General properties of the stationary states **

3.1 Introduction

3.2 General framework

3.2.1 The Cauchy problem

3.2.2 Global properties of the set of persistent states

3.3 Exploring the set of persistent states

3.3.1 A simpler case

3.3.2 Returning to the original equation

3.4 Reduction to a nite dimensional analysis

3.4.1 The Pincherle-Goursat Kernels

3.4.2 Persistent state equation for PG-kernels

3.4.3 Reduction to a nite number of ordinary dierential equations

3.5 One population of orientation tuned neurons: the Ring Model

3.5.1 Mapping the Ring Model to the PG-kernel formalism

3.5.2 Finding the persistent states

3.5.3 A closer inspection of contrast dependency and the broken symmetry

3.5.4 Discussion

3.6 Two populations of spatially organized neurons

3.6.1 Approximation of J

3.6.2 Numerical experiments

3.7 Discussion

3.7.1 Is the cortex really nite?

3.7.2 How steep should the sigmoid be?

3.8 Conclusion

**III Delayed neural eld equations **

**4 Theoretical propertie**s

4.1 Introduction

4.2 The neural eld model

4.3 Mathematical framework and notations

4.3.1 Solutions of the nonlinear problem

4.3.2 Boundedness of solutions in C

4.4 Linear analysis

4.4.1 Semigroup properties from the spectral study

4.4.2 Generalized eigenspaces

4.4.3 Spectral projector on generalized eigenspaces

4.4.4 Phase space decomposition

4.5 Stability results in C

4.5.1 Stability results in C from the characteristic values

4.5.2 Generalization of the model

4.5.3 Principle of the linear stability analysis via xed point theory in C

4.5.4 Summary of the dierent stability bounds

4.6 Center manifold reduction

4.6.1 Formulation as a Cauchy problem

4.6.2 Solution of the inhomogeneous problem

4.6.3 Center manifold and reduced equation

4.6.4 Normal form of the Pitchfork bifurcation

4.7 Conclusion

**5 Numerical and symbolic tools **

5.1 Numerical computations

5.1.1 Evolution equation

5.1.2 Spectrum computation

5.2 Symbolic computation of some normal forms

5.2.1 Maple program for computing the normal form N

5.2.2 Normal forms for 1D convolutional neural elds

5.3 Conclusion

6 Application to the study of dierent connectivities on a ring

6.1 Bifurcation analysis of two delayed neural eld equations

6.1.1 Inverted Mexican-hat connectivity

6.1.2 Mexican-hat connectivity

6.2 Numerical evaluation of the bounds in section 4.5.4

6.3 Conclusion

**7 The Deco-Roland model of long-range apparent motion perception**

7.1 Introduction

7.2 Neural eld model

7.3 Parameter tuning

7.4 Eect of communication delays and feedback

7.5 Eect of the intra-cortical propagation delays

7.6 Study of oscillatory patterns

7.7 Study of the apparent speed/contrast relationship

7.8 Conclusion

**IV Application to models study **

**8 General Introduction **

8.1 The mechanism of Ben-Yishai et al

8.2 The three models

**9 Illusory persistent states in the Ring Model of visual orientation selectivity **

9.1 Introduction

9.1.1 Chronology

9.1.2 Modelling with neural elds equations

9.1.3 Parametrization of the external input and of the connectivity function

9.1.4 Symmetries of the cortical network

9.1.5 General properties of the network and plan of the study

9.2 Reformulation of the problem and handling of the symmetries

9.2.1 Turning the problem into a nite dimensional one

9.2.2 Keeping only one mode in the connectivity J, N = 1

9.2.3 Keeping two modes in the connectivity J, N = 2

9.3 Tuning curves of the simplied equations

9.3.1 Finding the tuning curves, case N = 1

9.3.2 Finding the tuning curves, case N = 2

9.3.3 Arbitrary number of modes in the connectivity function

9.4 Dynamical 90 degrees illusory persistent state

9.5 Discussion

9.5.1 Parameter tuning

9.5.2 Perception threshold

9.5.3 Illusory persistent states

9.6 Conclusion

**10 The model of V1 by Blumenfeld et al. **

10.1 Introduction

10.2 Denition of the model parameters

10.3 Symmetry/statistical properties of the connectivity J

10.4 Simplication of the equations

10.5 Study of the spontaneous activity

10.6 Study of the evoked activity

10.7 Design of a dynamical stimulus

10.8 Conclusion

**11 A model of V1 without feature-based connectivity **

11.1 Introduction

11.2 A rate model with one population

11.2.1 The expression of the local connectivity, Jloc

11.2.2 The expression of the external input, Iext

11.2.3 The pinwheel lattice

11.3 Basic properties of the network: local connectivity

11.3.1 Network symmetries

11.3.2 Eigenvalue decomposition of the connectivity

11.3.3 A rst look at the spontaneous activity

11.4 Generalisation of the Ring Model

11.4.1 Identication of the critical wavevector

11.4.2 Parameter tuning and network behaviour

11.5 Space dependent delays eects

11.5.1 Case of constant delays

11.5.2 Case of space-dependent delays

11.6 Study of the long-range connections

11.7 Conclusion and extensions

**12 Conclusion **

Index

**V Appendix **

**A Stationary properties **

A.1 The center manifold theorem from [Haragus 2010]

A.2 Well-posedness of operators

A.3 Fixed points theorems

A.4 Compact operators with simple eigenvalues

A.5 Reduction of the activity based model to a nite number of ordinary dierential equations

A.6 Lemmas for the general bounds

A.7 The size of the basin of attraction of a stable persistent state

**B Theory of delays **

B.1 Operators and their spectra

B.2 Boundedness in F

B.3 Regularity

B.4 Analytical formula for the Hopf bifurcation curve in the parameter plane (D; c)

B.5 Study of the adjoint A

B.6 Linear analysis

B.6.1 Stability

B.7 The Cauchy problem

B.7.1 Regularity of R

B.7.2 The inhomogeneous equation

B.7.3 The inhomogeneous equation (Parabolic case)

**C Numerics of delays **

C.1 Denitions

C.2 Normal form computation

C.2.1 The Hopf bifurcation

C.2.2 Fold-Hopf bifurcation

C.2.3 Hopf-Hopf normal form

C.2.4 Hopf-Hopf bifurcation with modes n m

**D The Deco-Roland model of long range apparant motion in the ferret **

D.1 Eigenvalues and linearized equation

D.2 Computation of the bifurcation curves

D.3 Eect of the intra-cortical propagation delays

D.4 Study of the Hopf bifurcation curves for constant delays

D.5 Study of the apparent speed/contrast relationship

**E The Ring Model **

E.1 Numerical computation of the invariant functions

E.2 Numerical computation of the solutions of the nonlinear equations in the case N = 2

E.3 Existence of the Pitchfork bifurcation

E.4 The width of the tuning curves

E.5 Computation of the coecient 3

E.5.1 Computation of W

E.5.2 Finding 3

E.6 Stability of the tuning curves with external input turned on

**F A model of V1 without feature-based connectivity **

F.1 Lemmas for the bifurcation points

F.2 The case of the long-range connections

F.3 Computation of the D4-Pitchfork normal form

**Bibliography**