The goal of this workshop is to provide an overview of the current state of research in mathematical approaches to neuroscience. This vibrant area, seeded by successes in understanding nerve action potentials, dendritic processing, and the neural basis of EEG, has moved on to encompass increasingly sophisticated tools of modern applied mathematics. Included among these are Evans functions techniques for studying wave stability and bifurcation in tissue level models of synaptic and EEG activity, heteroclinic cycling in theories of olfactory coding, the use of geometric singular perturbation theory in understanding rhythmogenesis, stochastic differential equations describing inherent sources of neuronal noise, spike-density approaches to modelling network evolution, weakly nonlinear analysis of pattern formation, and the role of canards in organising neural dynamics. Importantly the workshop will also address the novel application of such techniques in two half-day sessions, one on audition and the other on Parkinsonian tremor and deep brain stimulation. Hence, participants will be drawn from both the mathematical and experimental sciences.
A further aim of this workshop will be to encourage other applied mathematicians into this thriving area of research where their work can have an impact on both experimental and computational neuroscience. Indeed a major challenge for the mathematical neuroscience community is to complement new biological understanding of network function with a mathematical understanding of dynamics for computation. In particular this will require studies that go beyond the mathematically tractable cases of highly symmetric and homogeneous networks and for us to understand the role that noise, inhomogeneities, delays, and feedback have to play in shaping the dynamic states of biological neural networks.
Mathematics of Parkinson's Disease and deep brain stimulation
Parkinson's disease (PD) is a well known degenerative disorder characterized primarily by motor manifestations, such as muscle rigidity, slowing or loss of movement, and resting tremor. While it is known that parkinsonism results from the degeneration of neurons that supply the brain with dopamine, surprisingly little else about PD has been firmly characterized, such that experimental investigation relating to PD remains a very active area of research.
Recent advances have highlighted an important role for mathematical approaches to complement the experimental study of PD. It has become clear that PD involves the breakdown of information flow in a network of neurons with a non-trivial connectivity architecture, with tremor in particular representing a spatiotemporally organized activity pattern that somehow emerges from the network under pathological conditions. The development and analysis of mathematical models for this network, harnessing the mathematical tools for studying the dynamics of coupled oscillator networks, may help to elucidate the mechanisms involved and to suggest corrective measures. On the therapeutic side, amazing success has recently been achieved by deep brain stimulation (DBS), in which an electrode is chronically implanted in a patient's brain, where it delivers high-frequency current pulses on a continual basis. Incredibly, although this procedure is routinely performed, the mechanisms for its effectiveness remain unknown; moreover, it requires the use of high stimulation levels and it only works for a subset of patients. Mathematical analysis has recently given new insight into how DBS works, and further analysis, such as a more detailed characterization of how various brain areas are impacted by stimulation and modeling of the effect of stimulation on network dynamics, offer hope for the derivation of improved, milder forms of DBS that work for larger segments of the patient population.
Mathematics of Hearing
The auditory system has attracted the attention of mathematical modelers for over four decades. The availability of much experimental data from a number of species (cat, gerbils, monkey etc...) and the ability to control the inputs to the auditory systems has made it very exciting to model. Consequently there has been a strong interaction between experiment and theory. Another reason for the attraction of this system is the wealth of new data on humans following cochlear implant procedures, where nerves are directly stimulated by electrodes receiving acoustic signals that have been processed by “mathematical algorithms”.
Nevertheless, these devices are still rather rudimentary, and patients are unable to decipher signals from competing sources (the source separation problem) or in noisy environments. These obstacles can only be surmounted by novel theoretical approaches to this system that will enable us to understand how brain nuclei process the information from the auditory nerve. Topics at the forefront of this research include the mathematics of neural signal detection, the impact of random neural activity (neural noise) on signal processing, the mathematical properties of the distinct cell populations the auditory nerve projects to, and the role played by the huge amount of feedback from higher brain centers back to earlier auditory brain centers. These issues will be addresses by the experts invited to this exciting thematic half-day.