Reddit mentions of Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (Computational Neuroscience) by Izhikevich, Eugene M. (2010) Paperback (Computational Neuroscience Series)

Going to jump in and take a stab at responding, if nobody minds...

Neuropsychology uses mathematics very rarely. Neuropsych is more about brain injury and rehabilitating the person around whatever neural issue they have. Neuropsychologists typically operate as part of a clinical treatment team, working alongside a neurologist, maybe a neurosurgeon if there was some intracranial or CNS trauma involved, and some sort of physical, behavioral, or cognitive therapist. In this team, neuropsychologists usually run the tests to diagnose disabilities or track symptoms over time. If you're interested in the neuroscience of psychology/cognition, you may be more interested in cognitive or behavioral neuroscience. These fields rely on mathematics but in a different way because the observations at that level are so hard to quantify. Mathematics in cognitive neuroscience (and even neuropsychology) is more about measurement theory: quantifying abstract or immeasurable phenomena and then attempting to explain how high-level function is tied to low-level events. Stuff that comes to mind includes the neurobiology of autism, visual attention, information processing in sensory networks, etc. This will lead into Bayesian decision theory, information theory, psychophysics, probability models, and from a very theoretical side, graph theory and looking at the mathematics of network topology and multi-sensory integration.

Mathematics is used in neurochemistry (or, more precisely, in fields like biochemistry, neuroendocrinology, neuropharmacology, biophysics, etc). In those fields, math is often used to describe the dynamics of whatever system you are studying, whether it's some kinetic process like diffusion or changes in protein conformation or receptor/chemical binding dynamics or even chemical metabolism. For this, you'll really want to know your differential equations and dynamical systems. The Dayan and Abbott textbook is great for this, but also look at http://www.amazon.com/Dynamical-Systems-Neuroscience-Excitability-Computational/dp/0262514206/ and even check out the journal Biological Cybernetics. Bertil Hille's book is also really good for things happening in and around the neuron.

I didn't find Theoretical Neuroscience particularly readable as others in the thread have said, but it is the go-to book for the classic topics in the field. I found Fundamentals of Computational Neuroscience to be a much much better book for introductions. From Computer to Brain : Foundations of Computational Neuroscience was fairly approachable. On the more cognitive side, From Neuron to Cognition via Computational Neuroscience was pretty good. If you like the nonlinear systems side, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting was pretty tough to read but full of good content.

It really depends on what subsets of comp neuro you're most interested in. I worked mostly on the cognitive side, and I was never super satisfied with any books on comp neuro in that area. I think the field is just too young for a great summary to exist beyond the neuronal/small network level.

There is a ton of interesting mathematics that goes into other areas of neuro that wouldn't typically be included in Computational Neuroscience. Different imaging methods, for instance, have some pretty fun math involved and are very active areas of research.

Izhikevich has a great book for mathematical descriptions of neurons and circuits, some cartoons and interpretable descriptions
https://www.amazon.com/Dynamical-Systems-Neuroscience-Excitability-Computational/dp/0262514206/ref=sr_1_1?ie=UTF8&qid=1543300294&sr=8-1&keywords=dynamical+systems+in+neuroscience
Edit: spelling

I can't advise on applying to PhD programs, but a book you should definitely look into is Dynamical Systems in Neuroscience by Izhikevich.

Slightly related, the intersection between neuroscience and topology will either by through ODEs (think phase plane analysis type stuff, which Izhikevich covers in detail) or analyzing neural networks via network analysis (so more graph theory type stuff) or algebraic topology (i.e. topological data analysis).

I think penn state has an active group doing neuroscience with topological tools. Specifically Carina Curto is someone to look into; if she doesn't do stuff you're particularly interested in, one of her collaborators likely will. She also wrote a survey article in the bulletin of the ams recently. I skimmed it and it looks like it will give a you a good flavour of how topology is used in neuroscience nowadays. Good luck!