It is natural to think about the brain and brain function on four different levels: genomics, biochemistry, electrophysiology, and behavior. Enormous amounts of new information are becoming available on associations between genotypes and behavior. The causal mechanisms, which are mostly unknown, necessarily involve the effects of genotype on development, cellular biochemistry, and electrophysiology. The cellular biochemistry and morphology of neurons is fundamental for understanding the electrophysiological properties of neurons and networks. And the network properties then give rise to the brain functions that we label with terms such as memory, mood, decision-making, motor control, and so forth. This simple characterization is misleading because the use of the word "level" suggests that there is bottom up control, the genes control the chemistry that controls the electrophysiology that controls behavior. The scientific issues are so difficult and interesting precisely because this is not true. Behavior affects gene expression levels, electrophysiology induces short and long term changes in cell biochemistry and morphology, which in turn influence the electrophysiology. On each of these levels, mathematicians and computational neuroscientists have created models to give conceptual understanding, to organize data, and to explore causal mechanisms. This workshop will focus on three particular areas. Morphology of neurons and electrophysiological processing. The great variety of dendritic morphologies suggest functional roles for different geometries and it is now understood that dendrites are often not passive conductors. Mathematical models have shown how the distribution of channels and receptor trafficking influence electrophysiological signaling. However, it is also known that electrophysiological signaling affects dendritic processing by affecting synapses and spines and other changes in morphology. For example, gonadotropin-releasing hormone cells of the hypothalamus drive the transition through puberty via changes in cellular- and population-level firing patterns. This change in electrical activity is accompanied by dendritic pruning that alters the electrical/conductance properties of the neuron. Medial superior olive (MSO) neurons in the auditory brainstem decrease their dendritic arborization during postnatal development, eventually achieving bipolar morphology. Mathematical models of the MSO and other neural populations suggest that not only the morphology but also the distribution of different ion channels contributes to dendritic computation. Mathematical models that relate cellular properties to the electrophysiology of neurons often raise new questions in deterministic and stochastic dynamical systems. These include the origins of mixed mode oscillations and bursting behavior, as well as the interplay of stochasticity and synchony.
From signaling molecules to behavior. The brain can be in different states with different corresponding behaviors. Signaling molecules play an important role in modulating state, and behavior interacts with the signaling molecules. For instance, the extracellular concentration of the neuromodulator adenosine, which increases during wakefulness and decreases during sleep, appears to increase propensity to transition from waking to sleep by inhibiting wake-active cholinergic cells. In turn, cholinergic cells play a role in inducing REM sleep as well as mediating cognitive functions via signaling on multiple time scales, and acetylcholine has long been recognized as a slow-acting neuromodulator of arousal states. While awake, behavioral activity can provide positive feedback helping to sustain wakefulness in the face of accumulating adenosine. Molecular and electrical signals interact to develop the neuronal network underlying the interactions described above. Moreover, cells have intrinsic mechanisms to tune properties of the electrical signal. For instance, cortical networks have mechanisms that facilitate context-dependent synchronization of different subnetworks within a fixed architecture. In other networks, such as the basal ganglia and spinal interneurons, ion channel mechanisms actively maintain a lack of correlation between nearby cells and this decorrelation may be important for executing smooth movements. Deterministic and stochastic dynamical systems models have been used to investigate the connections between molecular signaling and behavior.
Robustness and plasticity. An important property of brain function is that it must continue to operate at each of the four levels outlined in the introductory paragraph despite variation and change in the properties at individual levels. This is true within individuals where properties change as a function of time due to development, meals, emotional and environmental factors, and synapse and cell death. It is also important to understand why system properties are so similar between individuals despite great differences in local detail. And finally, assumptions about the stability of function across species form the basis for conducting experiments on animals and drawing conclusions about human brain function. These kinds of "homeostasis" questions occur in all biological systems, but they are particularly interesting and important in studying brain function for two reasons. First, since neurons are inherently sloppy and stochastic devices, there must be active processes at both the cellular and network level to reliably detect and sharpen electrophysiological signals in the stochastic and noisy environment. Second, one of the most important properties of brain function at all four levels is that it is flexible and changeable on both short and long time-scales. How can brain systems be controlled and homeostatic, yet flexible and changeable at the same time? The answers will be fundamental for understanding brain function and will likely require new advances in stochastic dynamical systems.