Mathematical and Theoretical Neuroscience - Cell, Network and Data Analysis

Preface

Contents

About the Authors

From Single Neuron Activity to Network Information Processing: Simulating Cortical Local Field Potentials and Thalamus Dynamic Regimes with Integrate-and-Fire Neurons

1 The Map and the Territory

2 Simulating Local Field Potential with Integrate and Fire Neurons

2.1 Problems and Solutions

2.2 Combining Integrate-and-Fire Neurons and Morphological Models

2.3 Combining IFN Networks and Morphological Simulations

3 Integrate and Fire Neurons Model of the Thalamus

3.1 Thalamic Neurons Modeling

3.2 Integrate-and-Fire Model of the Thalamus Reproduces Sleep/Wake Information Processing Transition

3.3 Perspectives

References

Computational Modeling as a Means to Defining Neuronal Spike Pattern Behaviors

1 Introduction

2 Computational Model of a Neuron

2.1 Neuro-computational Properties

2.2 Biophysically Meaningful Models

2.3 Integrate and Fire (IF) Models

2.4 Izhikevich Model

3 Spike Pattern Behaviors

4 Evolutionary Algorithm as a Tool for Modeling Neuronal Dynamics

4.1 Model Optimization Using the EA

4.2 Feature-Based Fitness Function

4.3 Fitness Landscape with a Feature Based Function

5 Modeling Spike Pattern Behaviors

5.1 Optimization Objectives with a Behavior

5.2 Parameter Space Exploration

6 Summary

References

Chemotactic Guidance of Growth Cones: A Hybrid Computational Model

1 Introduction

2 Methods

2.1 Evolution of Intracellular Chemical Fields Within the GC Domain

2.2 Computational Model of Axonal Outgrowth Guided by Chemotaxis

2.3 Quantitative Evaluation of Growth Cone Model Performance

3 Results

3.1 Diffusion-Driven Instability

3.2 In Silico Paths of Outgrowing Axons

3.3 Quantitative Assessment of the Axonal Chemoattractive Response

3.4 Quantitative Assessment of Axonal Outgrowth in Control Conditions

3.5 Qualitative Predictions of Axonal Counterintuitive Behaviours

4 Discussion

References

Mathematical Modelling of Cerebellar Granular Layer Neurons and Network Activity: Information Estimation, Population Behaviour and Robotic Abstractions

1 Introduction

2 Methods

2.1 Single Neuron Modeling

2.2 Cerebellar Granular Layer Information Processing

2.3 Model Based Methods for Hemodynamic Response

2.3.1 Balloon Model Based Prediction

2.3.2 Modified Windkessel Model Based Prediction

2.4 Evoked Local Field Potentials and Neural Mass Model

2.4.1 Cerebellum Granular Layer Neural Mass Model with Mossy Fibers Input Patterns

2.4.2 Reconstruction of Local Field Potential from Spiking Models

3 Spiking Neural Network Based on Cerebellum for Kinematics

4 Results

4.1 Estimation of MI at MF-GrC Relay

4.2 Variations in BOLD Response Measured Using Balloon Model and Modified Windkessel Model (MFWM)

4.3 Simulating Extracellular Potentials Recordings in Neural Mass Model (NMM) and Spiking Neural Network (SNN)

4.4 Optimized Kinematic Control Using SNN

5 Discussion

6 Conclusion

References

Bifurcation Analysis of a Sparse Neural Network with Cubic Topology

1 Introduction

2 Materials and Methods

3 Results

3.1 Primary Branch and Eigenvalues

3.1.1 Stationary Solutions

3.1.2 Limit-Point Bifurcations

3.1.3 Cusp Bifurcation

3.1.4 Branching-Point Bifurcations

3.2 Secondary Branches

3.2.1 Stationary Solutions

3.2.2 Limit-Point and Cusp Bifurcations

4 Discussion

References

Simultaneous Jumps in Interacting Particle Systems: From Neuronal Networks to a General Framework

1 Introduction

2 Mean Field Models in Neuroscience

2.1 Neuroscience Models with Simultaneous Jumps

3 A General Mean Field Model with Simultaneous Jumps

3.1 The Microscopic Dynamics

3.2 The Macroscopic Process

3.3 Assumptions on Coefficients

3.4 Propagation of Chaos and Rate of Convergence

References

Neural Fields: Localised States with Piece-Wise Constant Interactions

1 Introduction

2 Neural Fields in Circular Geometries: Top Hat Interactions

2.1 Construction

2.2 Stability

3 Discussion

Appendix: Circular Geometry for a Top Hat Kernel

References

Mathematical Models of Visual Perception Basedon Cortical Architectures

1 Introduction

2 The Mathematical Model

2.1 Lifting of the Stimulus in the Cortical Space

2.2 The Connectivity Kernels and the Affinity Matrix

2.3 Spectral Clustering and Perceptual Units

3 Numerical Simulations and Results

3.1 Numerical Approximations of the Kernels

3.2 Emergence of Percepts

4 Conclusions

References

Mathematical Models of Visual Perception for the Analysis of Geometrical Optical Illusions

1 Introduction

2 The Mathematical Model: Neurogeometry of the Primary Visual Cortex

2.1 The Set of Simple Cells Receptive Profiles

2.2 Output of Receptive Profiles

2.3 Hypercolumnar Structure

2.4 Cortical Connectivity

3 The Neuro-Mathematical Model for GOIs

3.1 Output of Simple Cells and Connectivity Metric

3.2 From Metric Tensor Field to Image Distortion

3.2.1 Strain Tensor: Displacement Vector Field

3.2.2 Poisson Problems: Displacement

4 Numerical Simulations and Results

4.1 Perceived Deformation in GOIs

5 Conclusion and Future Works

References

Exergaming for Autonomous Rehabilitation

1 Introduction

2 Methodology

3 Discussion

4 Conclusion

References

E-Infrastructures for Neuroscientists: The GAAIN and neuGRID Examples

1 Introduction

2 Methods

3 Results

4 Discussion

References

Theory and Application of Nonlinear Time Series Analysis

1 Introduction

2 Dynamical Systems

2.1 Attractors

2.2 Equivalence Class

3 Embedding of Time Series

4 Determination of Parameters for Phase Space Reconstruction

4.1 Lag Time

4.2 Embedding Dimension

5 Nonlinear Predictability

6 Geometrical and Dynamical Characterization of Attractors

7 Multivariate Time Series: Quantifying the Level of Interdependence

7.1 Cross Correlation

7.2 Mutual Information

7.3 Spearman Rank Coefficient

7.4 Slope Phase Coherence

8 Measures of Coupling Directionality

8.1 Granger Causality

8.2 Symbolic Transfer Entropy

9 Conclusions

References

Measures of Spike Train Synchrony and Directionality

1 Introduction

2 Measures of Spike Train Synchrony

2.1 Adaptive ISI-Distance

2.2 Adaptive SPIKE-Distance

2.3 Adaptive SPIKE-Synchronization

2.4 Selecting the Threshold Value

3 Measures of Spike Train Directionality

3.1 SPIKE-Order and Spike Train Order

3.2 Synfire Indicator

3.3 Statistical Significance

4 Outlook

References

Space-by-Time Tensor Decomposition for Single-Trial Analysis of Neural Signals

1 Introduction

2 Computational Framework Formulation

3 Space-by-Time Non-negative Matrix Factorization Algorithm

4 Variants of the Decomposition Algorithm

4.1 Orthogonality Constraints

4.2 Discrimination Objective

4.3 Application to Signed Data

5 Assessment of the Decompositions

5.1 Approximation Power of the Decomposition

5.2 Discrimination Power of the Decomposition

5.3 Model Order Selection

6 Example Application: Retinal Ganglion Cells

7 Software Implementation

8 Conclusions

References

Inverse Modeling for MEG/EEG Data

1 Introduction

2 Data Formation

3 The Inverse Problem

3.1 Classification of Inverse Methods

3.2 Methods for the Distributed Model

3.2.1 Minimum Norm Estimate (MNE)

3.2.2 Mixed Norm Estimates

3.2.3 Kalman Filtering

3.3 Methods for the Dipolar Model

3.3.1 Global Optimization Methods

3.3.2 Bayesian Monte Carlo Methods for Static Dipoles

3.3.3 Bayesian Monte Carlo Methods for Dynamic Dipoles

4 An Application to Epilepsy

5 Conclusions

References