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.
Author(s): Yoram Baram
Edition: 1
Publisher: World Scientific Publishing Company
Year: 2021
Language: English
Pages: 300
Tags: neural circuits; cognition; neuroscience;
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
