In the second decade of the twenty-first century, brain researchers (neuroscientists) have begun to decipher the dynamics of large-scale neural networks and to gauge how the functioning of the brain is dependent on the spatiotemporal integration of the resultant dynamics. Knowing more detailed facts about brain connectivity, we are faced with the problem of how such information can be put together in one system to understand the whole brain and the effective tracing of cause-and-effect relationships determining its actions. Data gathering by neuroinformaticians steadfastly produces a ‘brain in a supercomputer’ virtual model that can be regularly updated to fill in the missing pieces of the puzzle. Unfortunately, such data gathering does not imply that there is a “glue” that allows us to join together multiple empirical observations into a complete theory of the brain. In this respect, Mathematical Neuroscience provides a conceptual framework for integrating across scale. Firstly, the book considers the brain to be dynamically an open system with an infinite number of different kinds of problems. This requires an infinite system of equations to model an astronomical number of different neuronal patterns. Secondly, the book treats the dynamics as continuous in order to unify and explain the known behavior and physiological data across scale.
The prolific interconnectedness of the brain contrasts starkly with the brain as a multi-level/multi-scale dynamical system. Its complexity renders geometrical approaches in deciphering brain connectivity in ordinary 3D space limitless. While “multi-scale” modelers at distinct levels of neural organization impose bridges to approximate the continuity of dynamical systems across spatiotemporal scales leaving dynamical misalignments, Mathematical Neuroscience goes beyond such unrealistic dynamics of physical abstractions into the realm of integrative modeling and dynamic continuity. The book deals with continuity implicitly and provides an alternative to multi-scale modeling and geometric methods of analysis using tools of modern mathematics. Such a conceptual framework can crack open complex problems like the hard problem of consciousness that would be impossible to do on a computer. When it comes to solving the hard problem of consciousness, one must understand the issue of consciousness, being nowhere and everywhere in the brain, and thus not amenable to methodological reductionist analysis. Mathematical Neuroscience is the first book in mathematical neuroscience that can claim to include the development of a nonlinear analysis to better understand the intricate fallacies of methodological reductionism in neuroscience.
About Roman Poznanski
Roman R. Poznanski develops theories in neuroscience with mathematics. He recently co-authored the book Mathematical Neuroscience. His passion remains to pinpoint and crack open complex problems holding back our understanding of how the brain works. He is adamant, that the next Einstein in neuroscience will one day produce a complete theory of the brain, not as a hodgepodge of models, but as an integrative theory expressed in terms of modern mathematics. He has over 20 years experience as a theoretician and modeller in the neurosciences. He has edited several contemporary books: Biophysical Neural Networks (2001) and Modeling in the Neurosciences (1999, 2005). He is currently a visiting professor at the Rockefeller University.