Neural Control Engineering: The Emerging Intersection Between Control Theory and Neuroscience,

Steven Schiff’s *Neural Control Engineering: The Emerging Intersection Between Control Theory and Neuroscience* is largely concerned with predicting and controlling the dynamics of the brain. The problem involves collecting observations of brain activity and filtering out noise and measurement errors.

Issues related to brain dynamics are addressed in perhaps the most interesting part of the book—its final five chapters. Chapter 9 covers Apostolos Georgopoulos’s discovery in the 1980s that the direction of a monkey’s limb movement is uniquely predicted by the activity of a relatively small population of neurons in the motor cortex. Each of the neurons is tuned to respond optimally for a preferred direction of limb movement, and the vector sum of the neural population activity drives the intended movement. A corresponding effect occurs in the primate visual cortex: The vector sum of the activity of a relatively small population of visual-cortex neurons accurately represents the local orientation of the edge of an object in the visual field. Such findings were seminal for the development of brain–machine interfaces—for example, implanted arrays of microelectrodes. The resulting deluge of data generated by the arrays required much assimilation via so-called Kalman filters.

Chapter 10 provides an interesting introduction to Parkinson’s disease and the models developed at the turn of this century by David Terman and colleagues. Their approaches, using simplified Hodgkin–Huxley models, are a first attempt to model Parkinson’s disease and provide insight into the efficacy of deep brain stimulation. Chapters 11 and 12 give a brief look at the use of electric fields to stimulate the brain and at recent attempts to understand and control epileptic seizures. The final chapter, 13, is more speculative but raises the possibility that brains themselves implement Kalman filters.

Processing noisy data has its origins in Gauss’s 200-year-old least-squares method to minimize the effects of measurement errors. However, the current methods are offshoots of theories developed in the 1930s and 1940s by Norbert Wiener and Andrey Kolmogorov. Those theories minimized the effects of noise and measurement errors from data. Wiener’s theory considered continuously changing data that were represented with Gaussian statistics in stationary random processes; Kolmogorov studied very similar processes, but considered data sampled at discrete times.

The problem with both theories was that they dealt only with linear, stationary, and Gaussian processes. It took another 20 years before Rudolf Kálmán in 1960 introduced his filter theory to deal with noisy dynamical systems; it took another 40 years or so before Kalman filter techniques were used to estimate the parameters of equations describing neural activity.

*Neural Control Engineering* is the first comprehensive account of the most recent developments. Schiff is perhaps uniquely qualified to write it: He is a practicing neurosurgeon, a computational neuroscientist, and a pioneer in the application of control techniques to problems such as chaos. The book’s early chapters provide a brisk introduction to least-squares minimization and its connection with Bayes’s rule, and thence to processes that incorporate measurements into models of neural activity. In particular, Schiff presents the Kalman filter approach for discrete data, and the Kalman–Bucy filter for continuous data. The first specific neural examples the book considers are the Hodgkin–Huxley equations that model the ionic currents that trigger electrical pulses in neurons, and various simplifications, such as the FitzHugh–Nagumo equations. Schiff shows that such simplified models often lead to effective controls of neuronal activity.

Chapter 6 deals with a population model of large-scale neuronal activity, the Wilson–Cowan equations, which Hugh Wilson and I developed in the early 1970s. That model is essentially a spatiotemporal extension of equations like FitzHugh–Nagumo, and Schiff shows how a Kalman filter approach can be efficiently used to control the dynamics of circuits described by FitzHugh–Nagumo type equations. Chapters 7 and 8 deal with the construction of ab initio models and filters based directly on data assimilation and with model inadequacies. Chapter 7 discusses the utility of techniques that take data and abstract from it uncorrelated, linear sets that carry the essential information in the original data; those sets can then be used in a Kalman filter approach. Included examples illustrate applications to image analysis, both static and dynamic, and to the analysis of spatiotemporal brain activity.

I found *Neural Control Engineering* to be extremely interesting and well written. I have only two minor caveats. There is too little about the Wiener–Kolmogorov filters. And in chapter 11, the technique of reducing the resistive tree structure of a neural dendrite to an equivalent cylinder is introduced with no citation of Wilfrid Rall’s 1950s introduction of the method—Rall’s approach is based on impedance matching. Apart from those caveats, the book is a gold mine for anyone interested in learning how to model—and control—brain activity.