*Mathematical Foundations of Neuroscience* by G. Bard Ermentrout (University of Pittsburgh) and David H. Terman (Ohio State University) is volume 35 of Springer’s *Interdisciplinary and Applied Mathematics* series. This excellent 422 page hardcover publication is an accessible and concise monograph. Though not a textbook per se, *Mathematical Foundations* is a timely contribution that will prove useful to mathematics graduate students and faculty interested in the application of dynamical systems theory to cellular and systems neuroscience. The authors are card-carrying mathematicians with extensive research expertise in the field of theoretical neuroscience, numerous contributions to both the mathematics and scientific literature, and a long history of interdisciplinary curriculum development and instruction. Although not formally divided into two parts, chapters 1–7 of *Mathematical Foundations of Neuroscience* focus on cellular and subcellular aspects of neuronal modeling, while chapters 8–12 emphasize mathematical aspects of neuronal networks. The remainder of this review follows this conceptual division.

*Mathematical Foundations of Neuroscience* begins with a brief and conventional presentation of the equivalent circuit view of the plasma membrane. The current balance equation for the plasma membrane is then augmented with first order kinetic equations for the gating of two important ionic currents that result in the classical Hodgkin-Huxley equations for action potential exhibited by the squid giant axon. Subsequent introductory chapters discuss compartmental and cable modeling of the electronic properties of dendrites, phase-plane and bifurcation analysis of excitability and oscillations in the well-known Morris-Lecar model, and the dynamical consequences of several physiologically important ionic currents responsible for neuronal firing properties such as spike frequency adaptation and post-inhibitory rebound bursting. This introductory material is followed by chapters on mathematical aspects of electrical bursting, action potential propagation, and the dynamics of synaptic currents and plasticity.

This first portion of *Mathematical Foundations of Neuroscience* overlaps substantially with Eugene M. Izhikevich’s monograph *Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting* (MIT Press, 2007). A side-by-side comparison of the first half of Ermentrout-Terman with the whole of Izhikevich reveals that the latter includes more introductory biophysics and helpful exposition of membrane dynamics that can be observed in ordinary differential equation models with a single dependent variable, that is, Izhikevich “ramps up” more slowly than the first several chapters of Ermentrout-Terman. For this reason, as well as the high quality illustrations in Izhikevich, I would recommend that students unfamiliar with dynamical systems theory (such as undergraduate neuroscience majors) begin with *Dynamical Systems in Neuroscience* and then proceed to *Mathematical Foundations of Neuroscience,* which is written at a higher level and is significantly broader in scope. In particular, the single neuron modeling portion of *Mathematical Foundations of Neuroscience* includes important introductory topics such as cable modeling, singular construction of propagating action potentials, and synaptic dynamics that are not covered in *Dynamical Systems in Neuroscience.*

The second part of *Mathematical Foundations of Neuroscience* is a succinctly and balanced presentation of the dynamicist’s approach to systems (as opposed to cellular) neuroscience. Chapter 8 uses the analytical tools of phase response curves and averaging to explore the emergent dynamics of weakly coupled intrinsic neural oscillators (i.e., neuron models that when uncoupled fire repetitive action potentials). Chapter 9 analyses synchrony of coupled excitatory and inhibitory neurons, dynamic clustering, and propagating activity patterns in spatially distributed networks using “geometric singular perturbation theory,“ i.e., dissection of neuronal dynamics into fast and slow subsystems and the construction and analysis of singular systems composed of slow dynamics on (and jumps between) manifolds. While there is no detectable alternation of voice in these (my two favorite) chapters, one can imagine Ermentrout and Terman assigning these chapters to one another as they cover mathematical techniques each has championed.

The following chapter entitled “Noise” focuses on stochastic ordinary differential equation representation of synaptically driven leaky integrate-and-fire neurons and spike train statistics that can be derived by considering first passage times. Readers familiar with stochastic modeling might wish to supplement this chapter with selections from *Stochastic Methods in Neuroscience* (Carlo Laing and Gabriel Lord, editors, Oxford University Press, 2009). Readers unfamiliar with probability modeling might want to skim an elementary text such as Richard Durrett’s *Essentials of Stochastic Processes* (Springer, 1999) prior to reading this chapter. The organization of this chapter isn’t as logical as the rest of the book; in particular, the authors miss an opportunity to ground the Langevin description of membrane noise in the statistical properties of elementary processes such as the stochastic gating of ion channels or rates of excitatory and inhibitory synaptic input.

The final three chapters *Mathematical Foundations of Neuroscience* include discussions of population density methods, Wilson-Cowan-type and other “firing-rate” models of neuronal population activity, and spatially distributed neuronal networks that exhibit traveling wave solutions and stationary patterns. While the material presented in these chapters is well-organized and valuable, these topics are covered rather briefly. I found myself desiring one or two additional chapters focusing on the analysis of continuum “neural field” models where the strength of connection between neurons is represented by a synaptic footprint.

Taken as a whole, *Mathematical Foundations of Neuroscience* is a welcome addition to the pedagogical literature. Because of its emphasis on nonlinear dynamics and numerous examples of mathematical analysis of coupled neurons and neuronal networks, *Mathematical Foundations of Neuroscience* will appeal to mathematics graduate students more than complementary texts (Dayan and Abbott, 2005; Rieke et al.,1999) that emphasize the relationship between sensory stimuli and neural responses, information theoretic aspects of neural encoding and decoding, and so on. For mathematics graduate students who are investigating the field of computational neuroscience, I would highly recommend *Mathematical Foundations of Neuroscience* as their first computational neuroscience text.

Gregory D. Smith is an Associate Professor in the Department of Applied Science at The College of William and Mary, a Neuroscience Program Faculty Affiliate, and Director of William and Mary’s Biomathematics Initiative. His research interests include stochastic modeling in cell biology and neuroscience. He can be reached at greg@as.wm.edu.