Click on any of these titles to go directly to the corresponding section: To see publications that appeared during this period: Publications 1997


This year marked the 40th year of research in theoretical biology in the Mathematical Research Branch. Our work is carried out in close collaboration with experimentalists at NIH and elsewhere. This report describes work carried out during the 1997 fiscal year.

In addition to research activities, MRB members were involved in education through the hosting of two summer students, Mary Alderete of Arizona State University (through the MARC program), and Julie Cain from Penn State University-Erie. Also, A Sherman taught at the course, Methods in Computational Neuroscience in Woods Hole, MA, and J Rinzel taught at the Crete Course in Computational Neuroscience.

A Sherman helped organize the NIH Calcium Interest Group to further discussion and collaboration on calcium signalling among NIH institutes and other research institutes in the Washington-Baltimore area.

Cellular Neurobiology

Minimal models of respiratory bursting neurons in the Pre-Botzinger Complex: We have developed a minimal model of the oscillatory bursting neurons which are believed to generate the respiratory rhythm in neonatal rat brain slices. The model reproduces a number of experimentally observed properties, including a decrementing spike-frequency throughout each burst and burst frequencies which may vary from 0.1 to 1 Hz as intrinsic or extrinsic parameters are varied. Pairs of these neurons with excitatory (Glu) coupling were simulated to distinguish between the role of intrinsic and synaptic properties in controlling the burst cycle. Both extrinsic input and intrinsic properties may alter the frequency of the burst cycle. The excitatory coupling may prolong the length of the burst and extends the region of parameter space where other intrinsic parameters support synchronous bursting activity. Simulations of a population of heterogeneous bursting neurons reveals that heterogeneity of parameters is sufficient to account for the range of spike-frequency histograms (pre-I, synchronized I, and late I) recorded from neurons in neonatal brain slice preparations. (RJ Butera Jr, J Rinzel, and JC Smith: LNLC/NINDS).

Locomotion Pattern-Generating Properties of a Ring Circuit of Biophysical Neuron Models. A ring circuit of four neuronal oscillators produces patterns corresponding to several quadrupedal gaits, including the walk, bound, and gallop. An analysis using the phase-response curve (PRC) of the component neurons accurately predicted gait modes exhibited by the ring circuit and the phasic relationship between the component neurons. The patterns of network activity are determined by the PRC of the each neuron, which is determined by intrinsic biophysical properties. Altering the PRC of the neurons by a single intrinsic parameter was sufficient to produce gait transitions, with no change in the pattern or strength of connectivity among the neurons. Additional research has focused on predicting the stability of the patterns based on PRC analysis. (RJ Butera Jr, with RO Dror and JW Clark: Rice University, and CC Canavier, DA Baxter, and JH Byrne: University of Texas Medical School at Houston).

Spike frequency adaptation encourages synchrony in cortical networks: Oscillations in many cortical regions have common temporal characteristics. Experiments also reveal spatially synchronous oscillations which are stimulus dependent. This rhythmic activity suggests that coherence is a crucial feature of cortical networks. Using both simulations and a theoretical coupled oscillator approach, we demonstrate that spike frequency adaptation plays an important role in the dynamics of cortical networks. Without adaptation, excitatory connections among model pyramidal cells are desynchronizing; however, the slow processes associated with adaptation encourage stable synchronous behavior. (S Crook, J Bower: Caltech, and B Ermentrout: Pitt)

A case study for dendritic function: improving the performance of auditory coincidence detectors. Coincidence detector neurons in the auditory brainstem of mammals and birds have a known function in localizing sounds using interaural time differences (ITDs). Each neuron receives many narrow-band inputs from both ears and compares the time of arrival of the inputs with 10-100 \microsecond accuracy. Neurons that receive low frequency auditory inputs (up to about 2 kHz) have bipolar dendrites, and each dendrite receives inputs from only one ear. The correspondence between the known function of these neurons and their dendritic structure is not known. In a simple model that mimics the essence of the known electrophysiology and geometry of those cells, we show that dendrites improve the coincidence detection properties of the cells. The biophysical mechanism is based on nonlinear summation of excitatory inputs in each of the dendrites as well as using each dendrite as a current sink for inputs on the other dendrite. This is a rare case in which the contribution of dendrites to the known computation of a neuron may be understood. The results show that in these neurons, the cell morphology and the spatial distribution of the inputs enrich the computational power of these neurons beyond that expected from "point neurons". (H Agmon-Snir, J Rinzel, CE Carr: Univ. of Md.)

The shape of EPSP's affects the way in which they sum to trigger neuronal firing. Voltage-gated synaptic and membrane currents, in addition to passive effects of voltage on the synaptic current, can all alter EPSP shape as voltage changes. In order to disentangle these disparate mechanisms, synaptic responses in compartment models of layer 5 regular spiking cells were compared with experimental data collected in a slice preparation in which connected two pyramidal cells were simultaneously impaled. A systematic examination of the alternative mechanisms capable of reproducing the observed voltage dependencies of EPSP shape was then made. It was found that depolarization-gated EPSP broadening could be caused not only by synaptic NMDA currents but also by activation of several currents thought to exist in pyramidal dendrites, including the low threshold Ca2+ current CaT, the Na+ current, and also by inactivation of the transient K+ current IA. Testable predictions were made concerning the effect of blockers of these currents in slice, which have been confirmed in part by subsequently reported experimental work (Gilleson and Alzheimer 1997). It was concluded that a set of interacting voltage-gated currents participate in controlling EPSP shape in the subthreshold regime of synaptic integration in pyramidal cells. (P Rhodes and A Thomson: Royal Free Hospital, London)

It has become clear both that Na+ channels are present in the dendrites of pyramidal cells (Huguenard et al 1989) and that their presence alters the integration of synaptic inputs to these dendrites. Further, dendritic Na+ channels sustain the propagat ion of somatic firing into the dendritic tree, with consequences including the gating of Ca2+ entry by both membrane and synaptic channels. What has been less appreciated is that these phenomena are extraordinarily sensitive to the inactivation of the Na+ channel (Colbert and Johnston 1996). It is therefore of consequence that data from a variety of laboratories appear to disagree on parameters of inactivation (e.g. the voltage at which half inactivation is reached in steady state is -72 mV in Fleidervi sh et al 1996 versus -45 V in Cantrell et al 1995). These discrepancies may well be real, as studies of the channel itself (Isom et al 1994, 1995) reveal the existence of mechanisms to shift the gating of inactivation, including phosphorylation of the subunit and G-protein triggered effects mediated by the subunits of the fully constituted Na+ channel. In order to examine the functional consequences of shifts in the voltage gating of Na+ channel inactivation, a set of compartment model simulations was developed in which the integration of synaptic inputs to the basal and distal apical branches and ba ckpropagation of somatic firing into the dendrites were compared. It was found that just a 10 mV shift in the voltage at which inactivation occurs, from -55 to -65 mV, reduced excitability of the cell, reduced the occurrence of isolated dendritic spike events, and dampened the backpropagation of somatic firing into the dendritic tree. These results encourage the hypothesis that substances such as NE and ACh which appear capable of altering Na+ channel properties by the mechanisms noted above, may control the excitability of pyramidal cell dendritic tree in vivo. (P Rhodes)

The growing body of evidence demonstrating the presence of Na+, Ca2+, and K+ channels in pyramidal cell dendrites compels a re-examination of the functional properties of these cells. In particular, the integration of synaptic inputs in triggering firing, the interaction between inputs in the subthreshold regime, and the propagation of signals from the soma into the dendrites are examples of phenomena that are altered by voltage-gated dendritic currents. The wide variety of such channels and the complexity of their interaction with each other and with synaptic inputs suggests that the effects arising from their presence will be rich, and may alter our picture of pyramidal cell function. An invited review was prepared to gather together the evidence characterizing the channels present in pyramidal dendrites, and to review the functional consequences of dendritic currents in neocortical and hippocampal pyramdal cells. (P Rhodes)

Network Neurobiology

Multiple Coexisting Periodic Solutions in Modeled Bursting Neurons. Autonomous bursting systems are characterized by periods of repetitive activity punctuated by periods of silence. Recent studies have demonstrated that a model of neuron R15 may possess as many as 8 unique coexisting periodic solutions (multirhythmicity) at a given parameter set. This finding is of novel biological interest, where multirhythmicity has been proposed as a form of short-term memory, and of mathematical interest, since models with more than two coexisting periodic solutions are not frequently encountered. We developed a minimal biophysical model of neuron R15 and a mapping technique for identifying parameter sets where multirhythmic solutions occur. The origin of multirhythmicity in this class of models is hypothesized and demonstrated in a general 3 variable model, as well as a more complex biophysical model of neuron R15 with 11 state variables. It is possible that other bursting systems with at least two slow variables may exhibit similar forms of multirhythmicity (RJ Butera Jr).

Sensory Throughput of Thalamus: This long-term project is to address the circuit, intrinsic cellular, and modulatory properties in thalamus that contribute to throughput of visual sensory information. The work involves a joint experimental/computational collaboration. The experimental model is cat under in vivo and in vitro slice conditions, using pharmacological, electrophysiological, anatomical and imaging methodologies. Theoretical models are biophysically-based Hodgkin-Huxley-like neuron models in synaptically-coupled, spatially-distributed circuits. The collaboration of these two working groups (Sherman lab, experimental and Rinzel lab, computational) offers a new synergy of investigative experience of thalamic function in two distinct behavioral modes: the awake and asleep states. Most previous studies have focused on one or the other state and have been undertaken by groups backgrounded in one or the other. In general, this distinctness correlates with the widely-held view of thalamus exclusively as a simple linear relay for retinal information in the awake state and a nonlinear rhythm generator (pacemaker) for sleep states. Moreover, the two modes of behavior are consistent with the two different firing patterns seen at the level of individual cells (in both TC and TRN types): tonic spiking at depolarized levels ("aroused") and bursting at hyperpolarized levels ("sleep"). Recently emerging views, based on experimental evidence, however incorporate some alternate notions about thalamic sensory transmission such as it may not be simply feedforward or that it can be nonlinear with transient bursts perhaps signaling stimulus novelty. In recent years computational models have been developed (by Rinzel's group and others) that account for numerous rhythmogenic behaviors seen during sleep-like states, both in vitro and anesthetized in vivo. In these models intrathalamic circuitry (especially divergent feedback inhibition from TRN to TC) and the burst mode of firing play prominent roles in spationtemporal activity patterning. A major focus of the proposed work is to extend these biophysically-based models into the regime of the alert system and to consider, in-depth and systematically, whether the biophysical components that underlie sleep rhythms play any role during sensory processing. A hierarchy of models will be developed and experiments designed in order to bring in parallel the experimental and computational models from low arousal to alert states, and eventually to selective attention. (J Rinzel, SM Sherman and L Cox: SUNY Stony Brook, GD Smith)

Mutual synaptic inhibition can recruit and synchronize GABAergic neurons into collective activity in some networks, eg, thalamic slices and treated hippocampal slices. We have studied propagation and activity patterning in a network of locally connected model neurons that are coupled by GABA_ A synapses and that have postinhibitory rebound capability. A localized stimulus to the resting network can initiate activity that very slowly propagates by recruiting successive neighboring cells either in discrete groups, cyclically to give the wave a lurching appearance, or in a continuous manner. These inhibition-triggered wave patterns are in striking contrast to the familiar excitation-generated fast neuronal waves, and the very slow propagation speed depends critically upon the time constant for priming the rebound mechanism. In the wake of the recruitment wavefront, the network can return to rest (a solitary event) or show sustained oscillatory activity, depending on parameters and stimulus properties. The oscillatory firing patterns typically display asynchronous wavelike structures, with very different features depending on whether the network connectivity is on-center or off-center. If the shunting synapses are depolarizing rather than hyperpolarizing, such as with reversal potential 10-15 mV above rest, recruitment spreads smoothly and orders of magnitude faster. These synchronization and recruitment properties have implications generally for frequency control and spatio-temporal patterning in networks of GABAergic neurons, and particularly when shunting synapses may change sign in response to modulatory influences, such as during development or as driven by the circadian cycle. (J Rinzel, D Terman: Ohio State U, XJ Wang: Brandeis U, B Ermentrout: U Pittsburgh)

Spinal cord neuron populations execute synchronized rhythmic activity in isolated embryonic cords and in disinhibited cultures. The multimodal collective oscillations are infrequent episodic bursts of fast cycles (0.5 to 5 Hz). Synaptic coupling is primarily excitatory, and synaptic depression has been implicated as a possible rhythmogenic mechanism. We have developed a two-variable firing rate model in which cells spike tonically when stimulated. Synaptic depression accounts for the fast events - alternately allowing cells to mutually activate each other and then suppressing their interaction. By adding a slow modulatory variable, episodic behavior in the model can occur via several mechanisms. In association with this modeling effort we have also written a summary article for TINS on recent experimental and modeling work on synaptic depression and its significance for neural computation. (J Rinzel, W Senn: University Bern, and M O'Donovan, J Tabak: NINDS)

Secretory Physiology and Calcium Oscillations

Characterization of heterogeneous firing patterns in single beta cells: It has long been argued that single beta cells are incapable of bursting oscillations. Recent experiments have determined that single isolated mouse beta cells can burst, but that the cells show extensive heterogeneity. We have developed methods to collect statistics on mean active phase, silent phase and cycle duration. Based on these statistics, we have determined that the oscillatory behavior can be classified into three categories. We are in the process of determining whether there is a correlation between the different firing patterns and particular current-voltage relations. Finding such a correlation will help us gain an understanding of the mechanism of bursting in these cells. (L Satin and T Kinard: Medical College of Virginia, G de Vries, and A Sherman)

Different mechanisms for slow, medium, and fast bursting oscillations: Beta cells and islets are capable of a variety of oscillatory rhythms under different experimental conditions that can all be characterized as bursting. However, their periods range over two orders of magnitude, from a few seconds to a few minutes. We have begun to examine the hypothesis that these different bursting oscillations are due to distinct mechanisms. To demonstrate the feasibility of this hypothesis, we have imposed an independent oscillation of the conductance of the ATP-inhibited potassium channel on a standard model for islet bursting. By itself, this model is capable of producing both classic islet bursting (with a burst period of 15-60 seconds) and a faster, muscarinic burst rhythm (with a burst period as small as 5 seconds) in the presence of muscarinic agonists and glucose. With the imposed additional oscillation of the K_ATP conductance, the model is also capable of reproducing experimental observations of slow bursts (with a burst period of 1-4 minutes). We hypothesize that the origin of the oscillation in the K_ATP conductance could be metabolic in origin. (A Sherman, P Smolen: U. of Texas-Houston Med. Sch., and G de Vries)

Multiple bifurcations in a polynomial model of bursting oscillations: Bursting oscillations are commonly seen to be the primary mode of electrical behavior in a variety of nerve and endocrine cells. There are many mathematical models of bursting. We addressed the issue of being able to predict the type of bursting oscillation that can be produced by a model. A simplified model capable of exhibiting a wide variety of bursting oscillations was examined. By considering codimension-2 bifurcations associated with Hopf, homoclinic, and saddle-node of periodics bifurcations, a bifurcation map in two-dimensional parameter space was created. Each region on the map is characterized by qualitatively distinct bifurcation diagram and, hence, represents one type of bursting. The map elucidates the relationship between the various types of bursting oscillations. In addition, the map provides a different and broader view of the current classification scheme of bursting oscillations. (G de Vries)

Calcium Sparks: Punctate releases of Ca2+, called Ca2+ sparks, originate at the regular array of t-tubules in cardiac myocytes. We have developed a simple numerical model of Ca2+ spark formation and detection in this cell type. The model involves the numerical solution to a set of reaction-diffusion equations that incorporate Ca2+ release, cytosolic diffusion, and resequestration by SR Ca2+-ATPases, as well as the association and dissociation of Ca2+ with endogenous Ca2+-binding sites and the diffusible indicator dye fluo-3. The time-dependent spatial profile of fluorescence is then convoluted with a point spread function to simulate the contribution of out-of-focus signal. In its simplest version of the model, which assumes homogeneity and isotropy, local calcium flux equivalent of 2 pA lasting 10 ms reproduced the brightness and the time course of cardiac Ca2+ sparks. With the aid of the model simulations, we found that properties of the indicator, in particular its mobility, are by far the most significant determinants of the appearance of Ca2+ sparks (peak, half time of decay, and FWHM) whereas other Ca2+ handling mechanisms, e.g., the stationary buffers and pumps, are less influential. The underlying Ca2+ signal is more markedly confined in space and in time than the fluorescence signal is, and is significantly perturbed by submillimolar concentrations of the dye. Direct back calculating of Ca2+ from fluorescence, even use blur-free fluorescence data, fails to provide satisfactory results. This work indicates that a more reliable way to interpret fluorescence data is to directly model the fluorescence and to deduce the Ca2+ profile from the simulations. In more complex versions of the model, we showed that the asymmetric shape of Ca2+ spark (~20% longer in the longitudinal direction than in the transverse direction) is best explained by anisotropic diffusion of Ca2+ ions and the dye rather than by sub-sarcomeric inhomogeneities of the Ca2+ buffer and transport system. Further, effect of off-center confocal sampling on the apparent spark statistics was also examined. (GD Smith, JE Keizer, UC Davis, MD Stern: NIA, WJ Lederer: UMAB and H Cheng: NIA)

Spark-to-Wave Transition: During Ca2+ overload Ca2+ sparks serve as sites for the initiation of Ca2+ waves in myocytes. We have carried out computer simulations of spark-induced waves to explore the influence of the regular array of release sites on their propagation. The simulations combine Ca2+ diffusion with a simple kinetic model of the release site and a Ca2+ leak and reuptake into the SR. The kinetic model replicates the average rise and refractory times of a spark and includes adaptive behavior that mimics measurements on isolated RyRs in bilayers. Our computer simulations with a linear array of Ca2+ release sites give a wave speed proportional to the Ca2+ diffusion constant rather than its square root, as is true for reaction-diffusion equations in an excitable medium. We have developed a simplified ``fire-diffuse-fire'' model that mimics the properties of Ca2+-induced Ca2+ release from discrete release sites. This model suggests that continuous and saltatory Ca2+ waves may be differentiated macroscopically by the temperature- and Ca2+ buffer-dependence of their wave speeds. (JE Keizer: UC Davis, GD Smith, S Ponce-Dawson and J Pearson: Los Alamos)


The research has proceeded on the representation of the microcirculatory adjustments to different levels of tissue metabolic demands. The model includes 1. the microvasculature structure, the vascular wall Ca and K ion transports, the resulting dynamics of the terminal arteriole wall muscle (tone and vasomotion), and associated blood flow, 2. the effect that the tissue metabolic demands exert on the tissue oxygen and metabolites diffusion-consumption and resulting ADP, ATP and O2 concentrations, 3. the influence that the ADP and ATP concentration exerts on the vascular K-ATP and Ca channel conductance, respectively. The confluency of 1, 2 and 3 closes a metabolic-microcirculatory loop. The vasomotion response to physiologicaly increasing metabolic demands results in an increase of the blood flow that secures a minimal change on ADP and ATP tissue levels and avoids tissue hypoxia. J Gonzalez-Fernandez, A Sherman, B Ermentrout: Pitt)

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. 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; AQP1 is a water transporting channel protein expressed in proximal tubule and descending limb, and AQP2 is found in the collecting duct. Study of cirrhotic rats has shown down regulation of BSC-1 in the cortical thick ascending limb and of AQP2 in the collecting duct with up regulation of AQP1 and Na-K-ATPase in the proximal tubule. We have used a multinephron renal model to study the net effect upon the concentrating mechanism. A counterbalancing effect on urine osmolality, NaCl excretion and water excretion is predicted. (R Mejia, P Fernandez-Llama and M Knepper: NHLBI)

Concentration mechanism in the renal inner medulla. The mechanism by which the mammalian kidney concentrates solutes in the inner medulla has yet to be established, in particular since active solute transport is considered unlikely due to limited blood flow. We have developed models to test several hypotheses. These have shown that an elastic epithelial cell with two bathing solutions and distinct apical and basolateral transport properties will not concentrate fluid passively under peristaltic stress. We are presently developing a model of the inner medulla that includes both cellular and tubular compartments in order to investigate mechanisms further. (R Mejia, R Chadwick: NIDCD, H Layton: Duke U., B Schmidt-Nielsen: UFL, 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, based on domain decomposition methods, are being implemented to increase computational efficiency and reduce cpu time. (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, has been made available to investigators via the World Wide Web at gopher:// Since the NIH gopher site is expected to be terminated early in 1998, alternatives for availability of the database on the web are being considered. (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: U Arizona)

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