Annual Report 2008.pdf - SAMSI

- Sampling: Do data represent the entire network or are they based on only a subnetwork or
subgraph? This problem can be considered from both a sample designed based or a model based
perspective.
- Embeddability: Underlying existing dynamic network models is a continuous time stochastic
process even though the data used to study the models and their implications may come in the
form of repeated snapshots at discrete time points--a form of time sampling as opposed to node
sampling--or cumulative network links. Can we represent and estimate the continuous-time
parameters in the actual data realizations used to fit models?
- Prediction: In dynamic network settings, data generated over time there are a series of
forecasting problems. How should we evaluate alternative predictions from different models?
Possible working group leaders:
Alan Karr (NISS)
Possible participants:
David Banks (Stat, Duke), H. T. Banks (Math, North Carolina State), David Blei (Computer
Science, Princeton), Mark Handcock (Stat, Washington), Eric Kolaczyk (Math and Stat, Boston
University), Peter Mucha (Math, UNC), Jonathan Mattingly (Math, Duke)
2.2.5 Dynamics of Neuronal Activity
Dynamical models of the neuron date back to the pioneering work of Hodgkin-Huxley in
the 1950s. These models have contributed enormously to our basic understanding of the
functioning of nerve cells and their ability to produce and propagate an action potential,
synchronize between cells, inhibit firing and many other neuronal events. The analysis of these
models has benefited greatly from developments in dynamical systems that emphasize geometric
and qualitative features of models. A substantial community in applied mathematics and
biophysics has been involved in this research for many years.
More recently, statistical problems of fitting neuronal data have received considerable
attention. The recent work has emphasized use of point process methodology in analyzing
sequences of action potentials, or spike trains. There is now an identifiable sub-field of statistical
analysis of neuronal data, exemplified by three international workshops devoted to this subject.
(SAND3, the third workshop on Statistical Analysis of Neuronal Data, was held May 11-13,
2006, in Pittsburgh, with Emery Brown and Rob Kass as primary organizers.)
A SAMSI program in stochastic dynamics offers the ideal framework to bring together
mathematicians and statisticians from these scientific strands together. Moreover, the
brainstorming format of a SAMSI program is the perfect vehicle for finding common ground for
these two approaches. With the physical information that the dynamical models carry, they offer
the development of improved statistical models if they can be directly incorporated. For instance,
the shape of the action potential can be captured in a dynamical model and its influence on firing
could be used critically in the design of statistical models for the firing rate. On the other hand,
the deterministic models only reflect the reality of neuronal spiking in a qualitative way and the
refinement of the models based on an incorporation of data through statistical methods may lead
to a deeper understanding of the models and their range of applicability. This may occur through
parameter estimation, as a first step, or more complicated approaches to the assimilation of data
into the models and incorporation of noise effects based on systematic statistical procedures. A
further benefit may be the extension of statistical analyses to the context of multiple interacting
neurons and networks which have been studied extensively through dynamical models.