The Subcritical Brain: A Synergy of Segregated Neural Circuits in Memory, Cognition and Sensorimotor Control

Description

Have over a hundred years of brain research revealed all its secrets? This book is motivated by a realization that cortical structure and behavior can be explained by a synergy of seemingly different mathematical notions: global attractors, which define non-invertible neural firing rate dynamics, random graphs, which define connectivity of neural circuit, and prime numbers, which define the dimension and category of cortical operation. Quantum computation is shown to ratify the main conclusion of the book: loosely connected small neural circuits facilitate higher information storage and processing capacities than highly connected large circuits. While these essentially separate mathematical notions have not been commonly involved in the evolution of neuroscience, they are shown in this book to be strongly inter-related in the cortical arena. Furthermore, neurophysiological experiments, as well as observations of natural behavior and evidence found in medical testing of neurologically impaired patients, are shown to support, and to be supported by the mathematical findings. Contents Prologue Chapter 1. Introduction 1.1 General 1.2 Historical account 1.2.1 The nature of cortical information 1.2.2 Developmental aspects 1.2.3 Cortical connectivity 1.2.4 Somatic and synaptic polarization 1.2.5 Synaptic and somatic elimination 1.2.6 Cortical plasticity 1.2.7 Learning and memory 1.2.8 Cortical function 1.2.9 Graphs and categories 1.2.10 Prime numbers 1.2.11 Quantum computation 1.2.12 Sensorimotor control 1.3 Book outline Chapter 2. Some Mathematical Preliminaries 2.1 Introduction 2.2 Random graph criticality and connectivity 2.3 Primes, primality testing and prime factorization 2.4 Continuous and discrete-time dynamical systems 2.5 Elements of non-invertibility 2.6 Singularities of discrete iteration maps 2.7 Cobweb diagrams 2.8 From continuous-time to discrete-time dynamics of neural firing and plasticity 2.9 Self-regulation and stability in open and closed-loop control systems 2.10 Discussion Part I. Cortical Graphs and Neural Circuit Primes Chapter 3. Polarity Codes and Subcritical Linguistics 3.1 Introduction 3.2 Neuronal polarity codes 3.3 Subcritical neural circuit segregation 3.4 Hebbian linguistic impasse of probabilistically supercritical neural polarity 3.5 Critical Hebbian linguistic limit of neural polarity 3.6 Discussion Chapter 4. Hebbian Random Graphs and Firing-rate Dynamics 4.1 Introduction 4.2 Random Hebbian criticality effects on circuit linguistics 4.3 Firing-mode reproduction by polarity recall 4.4 Discussion Chapter 5. Synaptic Polarity and Primal-size Categories of Neural Circuit Codes 5.1 Introduction 5.2 Polarity codes and primal-size categories under axonal discharge self-feedback 5.3 Polarity codes and primal-size categories under synaptic self-feedback 5.4 Neural circuit polarity effects on firing dynamics 5.5 “Working Memory” and “Magical Numbers” 5.6 Discussion Chapter 6. Primal-size Neural Circuits in Trees of Meta-periodic Interaction 6.1 Introduction 6.2 Primal-size neural circuits in meta-periodic interaction 6.3 Primal-size circuit interaction in firing-rate meta-periodicity 6.4 Cortical information capacities under primal-size neural circuits 6.5 Primal-size neural circuits in meta-periodic trees 6.6 Discussion Part II. Firing-rate Linguistics Chapter 7. Firing-rate Dynamics Irreversibility 7.1 Introduction 7.2 Discrete iteration map of neural firing and synaptic plasticity 7.3 Non-invertibility (time-irreversibility) of the map 7.4 Discussion Chapter 8. Discrete Iteration Maps of Neural Firing in Cortical Development 8.1 Introduction 8.2 Reduced discrete iteration maps 8.3 Discrete iteration maps in developmental stages 8.3.1 Pre-critical period excitability 8.3.2 Critical period persistent plasticity 8.3.3 Convergent plasticity and synaptic maturity 8.3.4 Synaptic rigidity 8.4 Discussion Chapter 9. Global Attractors of Neural Firing-rate in Early Development 9.1 Introduction 9.2 Global attractors of neural firing-rate under pre-critical plasticity 9.3 Global attractors of neural firing-rate under critical plasticity 9.4 Robustness of firing-rate global attractors with respect to model parameter 9.5 Discussion Chapter 10. Global Attractors of Neural Firing-rate in Maturity 10.1 Introduction 10.2 Global attractors of neural firing-rate in maturity 10.3 Simulation results 10.4 Discussion Chapter 11. Firing-rate Mode Segregation by Neural Circuit Polarization 11.1 Introduction 11.2 Dynamics of somatic and synaptic silencing 11.3 Underlying asynchronous firing-rate and plasticity models 11.4 Asynchronous synaptic polarity-gated firing mode segregation 11.5 Firing mode control by synchronous polarity-gated circuit segregation 11.6 Neural circuit segregation capacity 11.7 Synaptic and somatic elimination: From pruning to senescence 11.8 Discussion Chapter 12. Learning and Memory by Circuit Polarization 12.1 Introduction 12.2 Short and long-term memory in the individual neuron 12.3 Short and long-term memory in synchronous circuit 12.4 Memory modification 12.5 Associative memory by circuit polarization and segregation 12.6 Discussion Part III. Cortical Quantum Effects Chapter 13. Some Quantum Computation Preliminaries 13.1 Introduction 13.2 States and qubits 13.3 Measurement 13.4 Operators and oracles 13.5 Grover’s quantum search algorithm 13.6 Discussion Chapter 14. Associative Memory by Quantum Set Intersection: The Edge of Small Neural Circuits 14.1 Introduction 14.2 Quantum set intersection 14.3 Quantum associative memory 14.3.1 Pattern completion 14.3.2 Pattern correction 14.4 Associative memory capacity results 14.4.1 Time complexity 14.4.2 Equilibrium storage capacity 14.4.3 Completion capacity 14.5 Numerical examples and simulations 14.6 Comparison to other works 14.7 Discussion Part IV. Sensorimotor Control Chapter 15. Circuit Polarity in Sensorimotor Control 15.1 Introduction 15.2 Obstacle detection 15.3 A bird’s eye view on the descent trajectory 15.4 Discussion Chapter 16. Autonomous Gait Entrainment in the Neurologically Impaired 16.1 Introduction 16.2 The edge of autonomy in visual feedback 16.3 The edge of auditory feedback 16.4 Dopamine reward and reward-seeking: The edge of memory 16.5 The edge of geometric dimension 16.6 The edge of polarity 16.7 Transformative autonomous entrainment of gait across neurological disorders 16.8 Discussion Chapter 17. Circuit Polarity and Singularity Segregation in Cortical Recordings 17.1 Introduction 17.2 Recorded cortical sensorimotor activity 17.3 Circuit polarization and singularity segregation in cortical recordings 17.4 Discussion Epilogue References Index