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To see a list of some of our publications click on this: Publications 1996


Current research projects of the Mathematical Research Branch reflect a range of interests in the development and application of theoretical models and of quantitative methodologies for understanding biological systems.

This research involves several different collaborations within the Branch and with other research groups, both at the NIH and elsewhere. This report describes recent work in the areas of oscillatory activity of secretory cells, cellular and network neurobiology, microcirculation, and renal physiology. During the past year, international collaborative projects have involved foreign investigators at Hebrew University, Jerusalem; at Ben-Gurion University, Beer-Sheva; University of Copenhagen; Royal Free Hospital, London; University of British Columbia, Vancouver; and Free University of Brussels.

Invited presentations were given by R Bertram (Gordon Conference on Theoretical Biology and Biomathematics), by J Rinzel (Symposium: Information Processing in the Visual Cortex and Beyond, Bad Honnef, Germany; Workshop: Nonlinear Dynamics of Networks of Neurons, Univ Calif-San Diego; Distinguished Lecture Series, Univ Wash; Distinguished Lectures on Computational Neuroscience, Univ Calif-Davis) and by R Mejia (Workshop: Minorities and Applied Mathematics-Connections to Industry, Institute for Mathematics and Its Applications, Univ Minn).

MRB staff were involved also with teaching activities. J Rinzel and A Sherman were invited to teach in the course, Methods in Computational Neuroscience, Woods Hole, MA. A Sherman was a principal lecturer in the Winter School in Mathematical Biology, University of Utah. J Rinzel lectured in the Crete Course in Computational Neuroscience.

Also during FY 96, A Sherman was granted tenure as an independent investigator in NIDDK. J Rinzel organized (with M Sherman, SUNY- Stony Brook and D Glanzman, NIMH) an international workshop: State-Dependent Function of Thalamus and Cortex, Washington DC.

Cellular Neurobiology

Coincidence detection mechanism in the auditory system. Using electrophysiological data gathered on cells in the auditory ITD (interaural time difference) pathway in birds, we developed a simple biophysical model for relay cells (in the N. magnocellularis) and coincidence detectors (in the N. laminaris). The simplest version of this Hodgkin-Huxley-like model assumes just two voltage-dependent currents, having just two dynamic variables and few parameters. The inward current produces the action potential's fast depolarization and the outward current repolarizes the cell back to rest. Input trains to the model mimic auditory nerve (AN) fiber activity for pure tone stimuli. Our analysis shows that the Poissonian property of the AN input train lowers the cell's phase-locking ability: the phase of the output spike depends on the interspike interval that precedes this spike (due to the refractory period after the previous output spike). This reduction in phase-locking can be avoided by using very large synaptic inputs. We also find that convergence of a few AN fibers onto a single relay cell may improve the phase-locking of the output of this cell. This improvement is found regardless of synaptic input amplitude (sub- or suprathreshold), and can be explained using concepts from order statistics. Generally, our present model fails to phase-lock and perform coincidence detection at high frequencies (>3kHz). However, our model gives insights on the mechanisms of phase locking and coincidence detection in these cells and on the problems found in the high frequency domain. (H Agmon-Snir, J Rinzel and C Carr: U Maryland)

Complex and bistable firing patterns in motoneurons: Recent experiments have determined that, in the presence of certain transmitters and ionic channel blockers, motoneurons generate complex and bistable firing patterns, such as calcium-based regenerative responses, bistable firing modes and plateau potentials. We investigate the roles of specific ionic currents and their distribution amongst the soma and dendrites in generating these firing patterns with a quantitative model containing experimentally determined conductances distributed between two compartments. This model is the first to account for a variety of experimentally observed motoneuron firing patterns and behaviors by simulating a systematic sequence of pharmacological experiments. We particularly focus on the effects of pharmacological agents that modulate the plateau-generating mechanisms but whose exact effects on ionic conductances are not known. Our simulation results predict that the agent serotonin can induce plateau potentials by modulating more than one ionic conductance and that plateau properties are sensitive to the degree of modulation. Of continuing interest is the mechanism underlying the observed slow activation of plateau potentials under current clamp. The biophysical conductances in the model generate slowly activating plateaus with only moderately large activation time constants, suggesting a novel interaction of relatively fast conductance activation and slower ionic concentration build-up in the slow generation of plateaus. (V Booth, J Rinzel and O Kiehn, Panum Institute, Copenhagen)

Modeling of thalamic relay cells: I noted that two postulated functions for thalamic relay cells, responding to "contextual" feedback from cortex and creating a reduced-redundancy representation of sensory input, could be implemented together if each thalamic relay cell received feedback which actively predicted the imminent sensory input, so that each relay cell would signal only the difference between sensory input and its prediction. I tested whether a simple model cell could implement such a "comparator" function by supplying random excitatory and inhibitory input to such a model, and reverse-correlating the cell's spikes and post-inhibitory-rebound bursts with the synaptic input which preceded them. My analysis showed that over a very wide range of input parameters, the cell model's spikes signaled an excess of excitatory input over inhibitory, and the bursts signaled an excess of inhibitory over excitatory input, thus producing two distinct types of signals which correspond quite well to the postulated comparator operations. (B Softky)

Dynamics of near-threshold bursting: In different types of bursting models, we investigate bursting oscillations near the threshold between steady and bursting behaviors. In square-wave bursting models, near-threshold bursts are characterized by an extended attraction to an unstable steady state that delays the initiation of the active phase of the burst trajectory. Using asymptotic methods, we obtain an estimate of the point of initiation of the active phase and a measure of the plateau fraction in this critical regime. In elliptic bursting models, we are investigating the mechanisms underlying the significant lengthening of the silent phase observed in near-threshold bursts. (V Booth, T Carr: Naval Research Laboratory, T Erneux: U Libre de Bruxelles, Belgium and ME Rush, Cal State U, Bakersfield)

Analysis of adaptation in cortical pyramidal cells: In cortical pyramidal neurons, the action potentials created by a sustained depolarizing current injection occur with a higher initial frequency and a decreased firing rate or a cessation of firing at later stages of the injection. This adaptation is mostly due to the activation of a slow calcium-dependent potassium current. We use a biophysical model of a pyramidal cell from piriform cortex to examine the underlying dynamics of the mechanisms that produce the adaptation. Our analysis shows that the degree of adaptation is determined by the ionic conductance density of the calcium-dependent potassium current and by the relative timing of the kinetics of this current and the rate of decay of intracellular calcium. (S Crook and B Ermentrout: U Pittsburgh)

Compartmental simulations of regular spiking neocortical pyramidal neurons: Under various manipulations (including high concentrations of EGTA) repetitive firing of regular spiking (RS) neocortical pyramidal neurons may be converted to a bursting mode. Based on previous modeling of intrinsically bursting (IB) pyramidal neurons, we hypothesized that RS cells may also have burst-mediating dendritic Ca currents, but that under ordinary conditions the dendritic Ca current and burst-capability is quenched by a [Ca]-gated K current. Using experimentally-determined morphology, we developed compartmental models of layer V RS cells. Our simulations support the above hypothesis: computed responses to somatic current injection under normal conditions show an adapting train of single spikes and, under sufficient EGTA, bursting is seen. As in model IB cells, the model RS cells have a dendritic distribution of high threshold Ca and also denditic Na channels, which are required to activate the high threshold Ca currents. (P Rhodes; R Yuste and D Tank: Bell Labs)

Neurotransmitter release and facilitation: A model of synaptic transmitter release and facilitation has been developed that is based on activation of release sites by single Ca++ microdomains centered at open Ca++ channels. Facilitation is due to Ca++ that remains bound to release sites between impulses. The model of release is inherently stochastic due to the stochastic opening and closing of Ca++ channels. However, it is sufficient to analyze the mean release, and we have derived deterministic equations that describe the time-evolution of this quantity. Mathematical analysis has led to an understanding of the many features of transmitter release and facilitation that are captured by the model. These include a release time course that is virtually independent of quantal content; facilitation without residual free Ca++; and steps in the frequency dependence of facilitation that correspond to a steplike decay in Ca++ cooperativity. Ongoing research is being performed to extend the model to longer-term forms of synaptic enhancement, including augmentation and post-tetanic potentiation. (A Sherman, R Bertram and E Stanley:NINDS)

Network Neurobiology

Compartmental simulations of a simple neocortical circuit: Only recently (Thomson et al 1988, 1989, 1995; Deuchars and Thomson 1994; Markram and Sakmann, in press) have dual impalement recordings of synaptically-connected pyramidal cells in cortical slice been possible. A collaborative project has been established to combine compartmental simulations of neocortical pyramidal cells with EPSP measurements from dual impalements in vitro. We are dissecting the mechanisms which could underlie the experimentally measured broadening of the EPSP with depolarization of the postsynaptic cell. Preliminary results suggest that not only synaptic NMDA current (which may play a smaller part than heretofore realized) but also subthreshold activation of dendritic and somatic Na, and low threshold Ca currents, along with inactivation of K(A), could contribute to the broadening. These features also impact the (synaptic) input-output properties of pyramidal neurons. A set of experimentally testable predictions for how pharmacological perturbations (e.g. intracellular block of Na channels with QX-314 or extracellular block of K(A) channels with 4-AP) might alter EPSP shape are being developed. (P Rhodes and A Thomson: Royal Free Hospital, London)

Prediction of sensory input: Following the hypothesis that neocortex learns to predict its sensory input, I constructed a network which uses Hebbian synaptic learning to predict the shape and position of a moving "stimulus." I invented a learning rule according to which synaptic strength changes in proportion to the difference between current predictions and to-be-predicted input, so that a stable state only exists when the prediction is optimal. And I verified that such a rule can learn tuning properties known to exist in visual cortex (i.e., edge detection, motion sensitivity, direction selectivity) directly from the properties of the input itself, without being "hard-wired." (B Softky)

Minimal biophysical models of oscillations and waves in thalamus and hippocampus: We describe, with minimal biophysical computational models, the dynamical behavior of seizure-like rhythms in thalamic and hippocampal slices. In the thalamus, we model the propagation of spindle waves in a network of excitatory thalamocortical and inhibitory reticular cells. As the wave advances, cells are recruited into the population rhythm, with reticular cells bursting almost every cycle at 7-10 Hz while thalamocortical cells burst only every few cycles. When GABA_A receptors are blocked all cells burst on nearly every cycle at 3-4 Hz, For the hippocampal slice we use a 2-compartment model for a CA3 neuron, a reduction of Traub's 19-compartment model. A one-dimensional network of hippocampal CA3 neuron models coupled by fast AMPA and slow NMDA synapses produces multiple synchronized population bursts. We distinguish between the discontinuous, "lurching" nature of the thalamic waves and the continuous nature of the hippocampal waves. (J Rinzel and D Golomb: Ben-Gurion Univ, Israel)

Experimentally-based model of sub-threshold oscillations in the inferior olive nucleus: We continue our theoretical modeling studies of the sub-threshold membrane potential oscillations (STO) seen in vitro in Inferior Olivary neurons. It appears that few if any individual cells spontaneously oscillate but that some slices where (presummably) there might be adequate connectivity (by electrical gap junctions) oscillate robustly. Thus we have formulated an heterogeniety hypothesis and have supported it by simulating the behavior of coupled cell pairs. We show that quiescent neurons can generate STOs when they are electrically coupled, provided that they differ in their ion channel densities. By numerical simulation of many different pairs, we have identified and developed a rule of thumb for the combinations of densities (for the two principal conductances, a transient calcium conductance and a leakage conductance) that lead to network rhythms. We have analytical results in limiting parameter regimes that support our "rule"; e.g., with strong coupling the arithmetic average of the cells' channel densities must be those of an "average" cell that is spontaneously oscillating. (J Rinzel, Y Manor: Brandeis Univ; I Segev and Y Yarom: Hebrew Univ, Israel).

Secretory Physiology and Calcium Oscillations

Biphasic response of pancreatic islets to glucose. The electrical response of pancreatic beta-cells to step increases in glucose concentration is biphasic, consisting of a prolonged phase a voltage spiking followed by voltage oscillations known as bursts. There is evidence that the first phase of continuous electrical activity is responsible for the first phase of elevated insulin secretion that is thought to be important for the normal regulation of blood glucose. We have successfully reproduced this biphasic glucose response using a mathematical model that we developed earlier to describe the effects of acetylcholine on beta-cell. In this model, Icrac, a current whose conductance as ER calcium decreases, plays a key role. I. Atwater and D. Mears (NIDDK) have confirmed using several distinct experimental maneuvers the prediction of the model that first phase duration depends on the filling state of the ER. For example, the duration increases with the length of time that the islet has been in low glucose , because the ER empties further. (A Sherman, R Bertram; I Atwater, E Rojas, and D Mears:LCBG/NIDDK)

Anti-phase, asymmetric and aperiodic oscillations in cells exhibiting bursting activity. We are examining the interaction of a pair of diffusively coupled biological bursters. Our motivation for this work is to understand the synchronization of bursting electrical activity observed in the beta cells of the islets of Langerhans. By studying the coupled fast subsystems of the bursters, using the slow variables as the control parameters, we have focused on the interaction that occurs during the active phase oscillations. When cells are very dissimilar and coupling strength is weak, we found the interaction can cause both bursters to stop oscillating. When cells are similar or coupling strength is relatively strong, they exhibit phase-locked, phase-trapped, or phase-drift behavior. The most prevalent behavior is the phase-locked behavior during which the spike amplitude is reduced compared to the uncoupled situation. Consequently, the burst period is prolonged, allowing for more calcium influx, which may be significant for insulin release. That is, weak coupling appears to be advantageous for the cells. (A Sherman and G de Vries)

Analysis of a class of models of bursting electrical activity in pancreatic beta cells. An experimental observation of bursting electrical activity in pancreatic beta cells is a correlation between the rate of insulin release from these cells and the plateau fraction (active phase duration as a percentage of the bursting period) as a function of glucose concentration. In a previous study of the Sherman-Rinzel-Keizer model of bursting electrical activity, Pernarowski developed analytical techniques to determine an approximation of the plateau fraction as a function of the glucose-dependent parameter a priori from the model equations. Applicability of these techniques depends critically on the fact that the fast subsystem of that model is an integrable system to leading order. We extended Pernarowski's results to a wide class of bursting models, namely those first-generation models consisting of three first-order ordinary differential equations. We showed that the fast subsystem of these models can be reformulated as an integrable system to leading order. The relative ease with which this reformulation can be done depends on a biological property of the models, namely the value of the integer exponent of the activation variable in the description of the voltage-gated potassium current. (G de Vries and RM Miura: Univ British Columbia, Vancouver).

Calcium signaling in pituitary-gonadotrophs. Based on our previous work on the two distinct calcium oscillators in pituitary-gonadotrophs, we further investigated the interaction between these two oscillators and the communication between plasma membrane and the endoplasmic reticulum. Based on available experimental data, our model incorporates spatially distributed cytoplasm and endoplasmic reticulum within a spherical cellular space bounded by the cell surface on which voltage-gated ionic channels and Ca2+ pumps are located. By assuming spherical symmetry in the distribution of channels and pumps, the 3D reaction-diffusion problem reduces to a 1D problem. We successfully explained the complex dynamic responses in the plasma membrane voltage, the cytoplasmic Ca2+ concentration and the ER Ca2+ concentration observed in gonadotrophs stimulated by the hypot halamic hormone GnRH. It also suggested plausible answers to problems of fundamental importance such as how the ER communicates with the plasma membrane in an excitable cell, and how calcium homeostasis is maintained before, during, and after agonist stimulation. In two invited reviews on recent progress in modeling agonist-induced calcium signaling, we addressed in depth the origin of calcium excitability. (Y-X Li, J Rinzel, J Keizer:UC Davis, S Stojilkovic:ERRB/NICHD)

Modeling NMDA-induced bursting in dopamine neurons and the basal ganglia-thalamocortical motor circuit. A model of the novel burst-generating mechanism in dopamine neurons of the basal ganglia was constructed and analyzed (Y-X Li, J Rinzel and R Bertram). Interaction between NMDA channels and Na+ pumps is crucial for the origin of the underlying slow rhythm. Efforts are now being made on constructing a model of the basal ganglia-thalamocortical motor circuit at the system level in order to study the role of dopamine in controlling the throughput of the basal ganglia by regulating the firing properties of striatal neurons. (Y-X Li and J Rinzel)


Metabolic control of vasomotion. It has been experimentally observed that the oscillations of arteriolar blood flow are related to the level of oxygen consumption; the duty cycle increases with oxygen consumption. The blood flow depends on the arteriolar lumen diameter, which in turn depends on the smooth muscle Ca concentration. The oxygen diffuses and is consumed in the tissues, resulting in a periarteriolar oxygen concentration, c. We have postulated that the resulting c modifies the conductance of K-ATP channels, analogous to metabolic control of oscillations in pancreatic beta-cells. We have also considered the alternative of modulation by ATP of voltage-dependent, L-type Ca channels. Either mechanism can produce vasomotion by controlling Ca entry into the smooth muscle and can reproduce the experimentally observed patterns of vasomotion. Also, the tissue oxygen concentration variation is reduced to only a small fraction of that seen when the dependence of the channels on c is abolished. (A Sherman, J Gonzalez-Fernandez and B Ermentrout: Univ Pittsburgh)

Structural Biology

Integral membrane proteins comprise receptors and transporters in epithelial cells that form the mammalian kidney. Neural net models that are trained using well-characterized proteins (e.g., G-protein coupled receptors) have been developed to identify the location of protein segments relative to the membrane or cytosole. Membrane segments are characterized as to membrane- or non-membrane-spanning and as to direction, into or out of the cell. Certain proteins (e.g., human oxytocin receptor, aquaporins, vasopressin-regulated urea transporter) have been characterized. The structure of coiled-coils, such as in synaptobrevin--syntaxin--SNAP-25 complexes, is being studied in an effort to develop criteria to characterize vesicle membrane fusion. (R Mejia and M Knepper: NHLBI)

Renal Physiology

Urine concentration mechanism. An electrical Na-K-2Cl transporter (BSC-1) is present in thick ascending limbs. Recent immunoblotting studies have shown that its abundance is increased by factors that raise glomerular filtration rate. We have used a multinephron model to show how these may have counterbalancing effects on salt excretion and maximum urine osmolality. (R Mejia and M Knepper: NHLBI)

Role of the kidney in acid/base balance. We have continued development of a multinephron model for acid/base balance in the whole kidney. Algorithms to increase computational efficiency and reduce cpu time required by whole kidney models are being implemented. (R Mejia and M Knepper: NHLBI)

Parameters for experimentalists and modelers. A database of renal parameters with author, title, abstract, address and source continues to be compiled. Physiologic and geometric parameters as well as conditions of measurement are included. This database, which is in extended MEDLINE format, is made available to investigators via the World Wide Web at gopher:// (R Mejia and M Knepper: NHLBI)

Cell Energetics

We have previously described the effect of changing transport work on the concentration profiles of high energy phosphate compounds within single cells using a reaction-diffusion model. A model with cylindrical geometry (pdes in two space dimensions and time) is used to investigate the biological hypothesis that ATP concentration is not limiting and that there can be significant ADP concentration gradients within the cell. (R Mejia and R Lynch: Univ Arizona)

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