Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems

Citation

Dayan, P., & Abbott, L. F. (2005). Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. The MIT Press.

Chapter Summary

The book has ten chapters in three parts, followed by a mathematical appendix and references. The authors' own description is that theoretical neuroscience gives a quantitative basis for describing what nervous systems do, determining how they function and uncovering the general principles by which they operate, with applications in vision, sensory-motor integration, development, learning and memory. Chapter titles below follow the authors' contents page.

Part I, Neural Encoding and Decoding: Chapters 1–4

Part I concerns the relationship between sensory stimuli and neural responses, with a focus on how the spiking activity of neurons represents information. The four chapters are "Neural encoding I: Firing rates and spike statistics", "Neural encoding II: Reverse correlation and visual receptive fields", "Neural decoding" and "Information theory". The first two describe neural responses statistically and relate them to stimuli, the third asks what can be read back from the responses, and the fourth supplies the information-theoretic framework for both questions.

Part II, Neurons and Neural Circuits: Chapters 5–7

Part II models neurons and circuits on the basis of cellular and synaptic biophysics. Chapter 5, "Model neurons I: Neuroelectronics", and Chapter 6, "Model neurons II: Conductances and morphology", build up single-neuron models. Chapter 7, "Network models", moves to circuits and is the chapter the authors have posted as a sample. It has sections on firing-rate models, feedforward networks, recurrent networks, excitatory-inhibitory networks and stochastic networks, and it works through coordinate transformations for visually guided reaching, selective amplification as a model for simple and complex cells in primary visual cortex, integration as short-term memory, gain modulation, associative memory, an olfactory-bulb model with oscillations, and the Boltzmann machine. The chapter ends with a summary, an appendix and an annotated bibliography.

Part III, Adaptation and Learning: Chapters 8–10

Part III analyses the role of plasticity in development and learning in three chapters: "Plasticity and learning", "Classical conditioning and reinforcement learning" and "Representational learning". The authors' contents page lists them without further description.

Mathematical appendix and references

The mathematical appendix collects the methods used throughout; Chapter 7 refers to it, for example, for the filtering of synaptic inputs. The complete reference list is also available on the authors' site.

Key Concepts

Part I: encoding and decoding

• Firing rates and spike statistics: describing a neuron's output as a rate and as a random process.
• Reverse correlation and receptive fields: estimating what stimulus feature drives a neuron.
• Neural decoding: estimating the stimulus from the response.
• Information theory: quantifying what responses convey about stimuli.

Part II: neurons and networks

• Neuroelectronics and conductances: modelling a neuron's membrane as an electrical circuit.
• Morphology: the effect of a neuron's shape on its responses.
• Firing-rate network models: feedforward and recurrent networks described by rates and weight matrices.
• Excitatory-inhibitory dynamics: oscillation and stability in coupled populations, analysed with linear stability analysis.
• Stochastic networks: probabilistic network models such as the Boltzmann machine.

Part III: plasticity and learning

• Synaptic plasticity: activity-dependent changes in synaptic strength as the basis of learning.
• Reinforcement learning: learning from reward, with classical conditioning as the example.
• Representational learning: how a network can learn internal representations of the statistics of its inputs.

Critical Analysis

Strengths

• Clear organisation. The three parts follow a single arc, from what neurons represent, through how neurons and circuits generate activity, to how circuits change with experience.
• Mix of analysis and simulation. The sample chapter shows the authors using analytical tools (weight matrices, eigenvectors, stability analysis, Lyapunov functions) and simulations side by side, with biological examples such as visual cortex and the olfactory bulb attached to each model.
• Chapter apparatus. Chapter 7 ends with a summary, an appendix of derivations and an annotated bibliography. The authors' site also offers exercises with code and data, figures for teaching, and an errata list for each printing.
• Authors' own resources. The free teaching materials make the book usable for self-study and for course design.

Limitations

• Mathematically demanding. The sample chapter relies on matrices and eigenvectors, differential equations, stability analysis and statistical ideas such as the Cramér–Rao bound, so readers need to be comfortable with these before starting.
• Dated in scope. The book was written in the early 2000s (the authors' errata page records two printings by February 2003), so it predates later developments in the field.
• Biology is assumed. The text is organised around the mathematics. Readers new to neuroscience will need a separate neuroscience textbook for the experimental background.

Where It Fits in the Course

This is the quantitative and theoretical book on the neuroscience list. Henley's Foundations of Neuroscience is a free introductory text for undergraduates new to the field, and Bear, Connors and Paradiso's Neuroscience: Exploring the Brain is an introductory textbook on the biological basis of behaviour. Purves et al.'s Neuroscience is a comprehensive textbook from single neurons to cognition, and Martin et al.'s From Neuron to Brain is a long-established cellular and systems text built on experiments. Luo's Principles of Neurobiology traces the experiments behind each principle.

Dayan and Abbott complement all of these: they take the biophysics and coding ideas that the textbooks describe in words and show how to express them as models. A student who has completed one of the neuroscience texts and the mathematics prerequisites can read the book next.

How to Use This Book

The book is for graduate students and advanced undergraduates in neuroscience, physics, engineering, mathematics and computer science who want to model the brain. Prior exposure to calculus, linear algebra, differential equations and probability is needed, and some programming helps with the exercises.

A practical plan:

• Core: Chapters 1–3 and 5–7 form the backbone: encoding, decoding, single-neuron models and networks.
• Choose by interest: Chapter 4 for information theory, Chapters 8–10 for learning and plasticity.
• Work the exercises. Use the authors' exercise sets and the accompanying code and data. Try Chapter 7 first if you want to judge the level before buying: the chapter is free on the authors' site.
• Check the errata for the printing you own.