Networks of nonlinear systems contain unknown parameters and dynamical degrees of freedom that may not be observable with existing instruments. From observable state variables, we want to estimate the connectivity of a model of such a network and determine the full state of the model at the termination of a temporal observation window during which measurements transfer information to a model of the network. The model state at the termination of a measurement window acts as an initial condition for predicting the future behavior of the network. This allows the validation (or invalidation) of the model as a representation of the dynamical processes producing the observations. Once the model has been tested against new data, it may be utilized as a predictor of responses to innovative stimuli or forcing. We describe a general framework for the tasks involved in the "inverse" problem of determining properties of a model built to represent measured output from physical, biological, or other processes when the measurements are noisy, the model has errors, and the state of the model is unknown when measurements begin. This framework is called statistical data assimilation and is the best one can do in estimating model properties through the use of the conditional probability distributions of the model state variables, conditioned on observations. There is a very broad arena of applications of the methods described. These include numerical weather prediction, properties of nonlinear electrical circuitry, and determining the biophysical properties of functional networks of neurons. Illustrative examples will be given of (1) estimating the connectivity among neurons with known dynamics in a network of unknown connectivity, and (2) estimating the biophysical properties of individual neurons in vitro taken from a functional network underlying vocalization in songbirds.

Representing the biophysics of neuronal dynamics and behavior offers a principled analysis-by-synthesis approach toward understanding mechanisms of nervous system functions. We report on a set of procedures assimilating and emulating neurobiological data on a neuromorphic very large scale integrated (VLSI) circuit. The analog VLSI chip, NeuroDyn, features 384 digitally programmable parameters specifying for 4 generalized Hodgkin-Huxley neurons coupled through 12 conductance-based chemical synapses. The parameters also describe reversal potentials, maximal conductances, and spline regressed kinetic functions for ion channel gating variables. In one set of experiments, we assimilated membrane potential recorded from one of the neurons on the chip to the model structure upon which NeuroDyn was designed using the known current input sequence. We arrived at the programmed parameters except for model errors due to analog imperfections in the chip fabrication. In a related set of experiments, we replicated songbird individual neuron dynamics on NeuroDyn by estimating and configuring parameters extracted using data assimilation from intracellular neural recordings. Faithful emulation of detailed biophysical neural dynamics will enable the use of NeuroDyn as a tool to probe electrical and molecular properties of functional neural circuits. Neuroscience applications include studying the relationship between molecular properties of neurons and the emergence of different spike patterns or different brain behaviors. Clinical applications include studying and predicting effects of neuromodulators or neurodegenerative diseases on ion channel kinetics.

With the goal of building a model of the HVC nucleus in the avian song system, we discuss in detail a model of HVC _RA projection neurons comprised of a somatic compartment with fast Na ^+ and K ^+ currents and a dendritic compartment with slower Ca ^2+ dynamics. We show this model qualitatively exhibits many observed electrophysiological behaviors. We then show in numerical procedures how one can design and analyze feasible laboratory experiments that allow the estimation of all of the many parameters and unmeasured dynamical variables, given observations of the somatic voltage V_s(t) alone. A key to this procedure is to initially estimate the slow dynamics associated with Ca, blocking the fast Na and K variations, and then with the Ca parameters fixed estimate the fast Na and K dynamics. This separation of time scales provides a numerically robust method for completing the full neuron model, and the efficacy of the method is tested by prediction when observations are complete. The simulation provides a framework for the slice preparation experiments and illustrates the use of data assimilation methods for the design of those experiments.

Data assimilation (DA) solves the inverse problem of inferring initial conditions given data and a model. Here we use biophysically motivated HodgkinHuxley (HH) models of avian HVCI neurons, experimentally obtained recordings of these neurons, and our data assimilation algorithm to infer the full set of parameters and a minimal set of ionic currents precisely reproducing the observed waveform information. We find many distinct validated sets of parameters selected by our DA method and choice of model. We conclude exploring variations on the inverse problem applied to neurons producing accurate or inaccurate results; by manipulating data presented to the algorithm, varying sample rate and waveform; and by manipulating the model by adding and subtracting ionic currents.

Techniques of data assimilation (DA) for parameter identification and forecasting in complex dynamical systems offer promising tools for the analysis of neural data and inference of neural function. Here we present experiments using DA to characterize the dynamics of a neuromorphic very large-scale integrated (VLSI) circuit emulating membrane dynamics and channel kinetics in a network of 4 generalized Hodgkin-Huxley neurons coupled through 12 conductance-based chemical synapses. The analog VLSI chip, NeuroDyn, features 384 digitally programmable parameters specifying for all neurons and synapses reversal potentials, conductances, and spline-regressed voltage dependence profile of opening and closing rates of the gating variables. In a set of experiments, we conducted DA on the membrane potentials of neurons recorded on the chip under current injection according to the model structure upon which the chip was designed and the known current input sequence, to arrive at the programmed parameters save for model errors due to analog imperfections in the chip fabrication. In a related set of experiments we extended the DA procedure to map songbird neural dynamics onto the chip by identifying and programming parameters extracted using DA from intracellular neural recordings. Application of the chip to neurological data may help to understand the effects of neuromodulators or neurodegenerative diseases on ion channel kinetics, and may further provide insights into the relationship between molecular properties of neurons and the emergence of different spike patterns or different brain behaviors.