Mathematical and Theoretical Neuroscience (eBook)

Prof. Giovanni Naldi studied Mathematics at the University of Pavia and at the University of Milan, where he also received his PhD in Applied Mathematics. He is currently a full professor of Numerical Analysis at the University of Milan and the director of the ADAMSS (ADvanced Applied Mathematical and Statistical Sciences) Center at the same University. His research work mainly focuses on the numerical analysis of partial differential equations, wavelet-based methods; multiscale models, non-linear evolution phenomena, biomathematics, and computational neuroscience. He has supervised eight doctoral theses and is the author of more than 60 papers. Prof. Thiery Nieus received his PhD in Applied Mathematics at the Department of Mathematics F. Enriques in Milan (Italy). His research focuses on the computations performed by neuronal networks. His work involves the analysis and modeling of multiscale data, ranging from single synapses to population recordings. In September 2016 he joined Marcello Massimini’s laboratory (University of Milan, Italy), working on computational models of the thalamocortical circuit and on complexity measures of TMS/EEG data.

Preface 6
Contents 8
About the Authors 10
From Single Neuron Activity to Network Information Processing: Simulating Cortical Local Field Potentials and Thalamus Dynamic Regimes with Integrate-and-Fire Neurons 11
1 The Map and the Territory 12
2 Simulating Local Field Potential with Integrate and Fire Neurons 14
2.1 Problems and Solutions 14
2.2 Combining Integrate-and-Fire Neurons and Morphological Models 15
2.3 Combining IFN Networks and Morphological Simulations 18
3 Integrate and Fire Neurons Model of the Thalamus 23
3.1 Thalamic Neurons Modeling 26
3.2 Integrate-and-Fire Model of the Thalamus Reproduces Sleep/Wake Information Processing Transition 27
3.3 Perspectives 30
References 31
Computational Modeling as a Means to Defining Neuronal Spike Pattern Behaviors 34
1 Introduction 34
2 Computational Model of a Neuron 35
2.1 Neuro-computational Properties 36
2.2 Biophysically Meaningful Models 37
2.3 Integrate and Fire (IF) Models 38
2.4 Izhikevich Model 39
3 Spike Pattern Behaviors 40
4 Evolutionary Algorithm as a Tool for Modeling Neuronal Dynamics 42
4.1 Model Optimization Using the EA 42
4.2 Feature-Based Fitness Function 43
4.3 Fitness Landscape with a Feature Based Function 44
5 Modeling Spike Pattern Behaviors 47
5.1 Optimization Objectives with a Behavior 47
5.2 Parameter Space Exploration 48
6 Summary 50
References 50
Chemotactic Guidance of Growth Cones: A Hybrid Computational Model 53
1 Introduction 54
2 Methods 54
2.1 Evolution of Intracellular Chemical Fields Within the GC Domain 54
2.2 Computational Model of Axonal Outgrowth Guided by Chemotaxis 56
2.3 Quantitative Evaluation of Growth Cone Model Performance 58
3 Results 59
3.1 Diffusion-Driven Instability 59
3.2 In Silico Paths of Outgrowing Axons 59
3.3 Quantitative Assessment of the Axonal Chemoattractive Response 60
3.4 Quantitative Assessment of Axonal Outgrowth in Control Conditions 62
3.5 Qualitative Predictions of Axonal Counterintuitive Behaviours 62
4 Discussion 64
References 65
Mathematical Modelling of Cerebellar Granular Layer Neurons and Network Activity: Information Estimation, Population Behaviour and Robotic Abstractions 68
1 Introduction 68
2 Methods 71
2.1 Single Neuron Modeling 71
2.2 Cerebellar Granular Layer Information Processing 73
2.3 Model Based Methods for Hemodynamic Response 74
2.3.1 Balloon Model Based Prediction 75
2.3.2 Modified Windkessel Model Based Prediction 76
2.4 Evoked Local Field Potentials and Neural Mass Model 77
2.4.1 Cerebellum Granular Layer Neural Mass Model with Mossy Fibers Input Patterns 77
2.4.2 Reconstruction of Local Field Potential from Spiking Models 78
3 Spiking Neural Network Based on Cerebellum for Kinematics 79
4 Results 80
4.1 Estimation of MI at MF-GrC Relay 80
4.2 Variations in BOLD Response Measured Using Balloon Model and Modified Windkessel Model (MFWM) 82
4.3 Simulating Extracellular Potentials Recordings in Neural Mass Model (NMM) and Spiking Neural Network (SNN) 83
4.4 Optimized Kinematic Control Using SNN 84
5 Discussion 86
6 Conclusion 88
References 88
Bifurcation Analysis of a Sparse Neural Network with Cubic Topology 93
1 Introduction 93
2 Materials and Methods 95
3 Results 96
3.1 Primary Branch and Eigenvalues 96
3.1.1 Stationary Solutions 96
3.1.2 Limit-Point Bifurcations 98
3.1.3 Cusp Bifurcation 98
3.1.4 Branching-Point Bifurcations 99
3.2 Secondary Branches 100
3.2.1 Stationary Solutions 100
3.2.2 Limit-Point and Cusp Bifurcations 102
4 Discussion 102
References 103
Simultaneous Jumps in Interacting Particle Systems: From Neuronal Networks to a General Framework 105
1 Introduction 105
2 Mean Field Models in Neuroscience 106
2.1 Neuroscience Models with Simultaneous Jumps 106
3 A General Mean Field Model with Simultaneous Jumps 108
3.1 The Microscopic Dynamics 108
3.2 The Macroscopic Process 111
3.3 Assumptions on Coefficients 112
3.4 Propagation of Chaos and Rate of Convergence 114
References 115
Neural Fields: Localised States with Piece-Wise Constant Interactions 117
1 Introduction 117
2 Neural Fields in Circular Geometries: Top Hat Interactions 119
2.1 Construction 120
2.2 Stability 122
3 Discussion 124
Appendix: Circular Geometry for a Top Hat Kernel 124
References 126
Mathematical Models of Visual Perception Basedon Cortical Architectures 128
1 Introduction 128
2 The Mathematical Model 130
2.1 Lifting of the Stimulus in the Cortical Space 130
2.2 The Connectivity Kernels and the Affinity Matrix 130
2.3 Spectral Clustering and Perceptual Units 134
3 Numerical Simulations and Results 134
3.1 Numerical Approximations of the Kernels 134
3.2 Emergence of Percepts 135
4 Conclusions 137
References 137
Mathematical Models of Visual Perception for the Analysis of Geometrical Optical Illusions 139
1 Introduction 140
2 The Mathematical Model: Neurogeometry of the Primary Visual Cortex 142
2.1 The Set of Simple Cells Receptive Profiles 142
2.2 Output of Receptive Profiles 142
2.3 Hypercolumnar Structure 143
2.4 Cortical Connectivity 144
3 The Neuro-Mathematical Model for GOIs 145
3.1 Output of Simple Cells and Connectivity Metric 145
3.2 From Metric Tensor Field to Image Distortion 147
3.2.1 Strain Tensor: Displacement Vector Field 147
3.2.2 Poisson Problems: Displacement 148
4 Numerical Simulations and Results 149
4.1 Perceived Deformation in GOIs 149
5 Conclusion and Future Works 151
References 152
Exergaming for Autonomous Rehabilitation 154
1 Introduction 154
2 Methodology 155
3 Discussion 160
4 Conclusion 162
References 162
E-Infrastructures for Neuroscientists: The GAAIN and neuGRID Examples 164
1 Introduction 165
2 Methods 166
3 Results 175
4 Discussion 176
References 177
Theory and Application of Nonlinear Time Series Analysis 180
1 Introduction 180
2 Dynamical Systems 181
2.1 Attractors 181
2.2 Equivalence Class 183
3 Embedding of Time Series 184
4 Determination of Parameters for Phase Space Reconstruction 186
4.1 Lag Time 186
4.2 Embedding Dimension 188
5 Nonlinear Predictability 190
6 Geometrical and Dynamical Characterization of Attractors 193
7 Multivariate Time Series: Quantifying the Level of Interdependence 194
7.1 Cross Correlation 195
7.2 Mutual Information 195
7.3 Spearman Rank Coefficient 196
7.4 Slope Phase Coherence 198
8 Measures of Coupling Directionality 198
8.1 Granger Causality 199
8.2 Symbolic Transfer Entropy 200
9 Conclusions 201
References 202
Measures of Spike Train Synchrony and Directionality 204
1 Introduction 205
2 Measures of Spike Train Synchrony 206
2.1 Adaptive ISI-Distance 207
2.2 Adaptive SPIKE-Distance 208
2.3 Adaptive SPIKE-Synchronization 210
2.4 Selecting the Threshold Value 214
3 Measures of Spike Train Directionality 215
3.1 SPIKE-Order and Spike Train Order 217
3.2 Synfire Indicator 220
3.3 Statistical Significance 222
4 Outlook 223
References 224
Space-by-Time Tensor Decomposition for Single-Trial Analysis of Neural Signals 226
1 Introduction 226
2 Computational Framework Formulation 228
3 Space-by-Time Non-negative Matrix Factorization Algorithm 229
4 Variants of the Decomposition Algorithm 230
4.1 Orthogonality Constraints 231
4.2 Discrimination Objective 231
4.3 Application to Signed Data 232
5 Assessment of the Decompositions 235
5.1 Approximation Power of the Decomposition 235
5.2 Discrimination Power of the Decomposition 235
5.3 Model Order Selection 236
6 Example Application: Retinal Ganglion Cells 236
7 Software Implementation 238
8 Conclusions 238
References 239
Inverse Modeling for MEG/EEG Data 241
1 Introduction 241
2 Data Formation 243
3 The Inverse Problem 245
3.1 Classification of Inverse Methods 247
3.2 Methods for the Distributed Model 248
3.2.1 Minimum Norm Estimate (MNE) 248
3.2.2 Mixed Norm Estimates 248
3.2.3 Kalman Filtering 249
3.3 Methods for the Dipolar Model 250
3.3.1 Global Optimization Methods 250
3.3.2 Bayesian Monte Carlo Methods for Static Dipoles 250
3.3.3 Bayesian Monte Carlo Methods for Dynamic Dipoles 251
4 An Application to Epilepsy 251
5 Conclusions 252
References 253