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To see a list of some of our publications click on this: Publications 1994

Introduction to MRB Research

Current research projects of the Mathematical Research Branch reflect a broad 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 cellular and network neurobiology, oscillatory activity of secretory cells, microcirculation, and renal physiology. During the past year, international collaborative projects have involved foreign investigators at Hebrew University, Jerusalem (Department of Neurobiology), and at the University of Alicante, Alicante, Spain.

Invited presentations were given at distinguished symosia by A Sherman (Pancreatic Islet Study Group of the European Association for the Study of Diabetes, Alicante, Spain), J Rinzel (International Hebb Symposium, Toronto; International Neural Network Society Satellite Symposium to Annual Meeting, Society for Neuroscience, Washington DC), Y-X Li (Gordon Research Conference on Theoretical Biology and Biomathematics, Tilton NH; Workshop on Calcium Waves and Dynamics, Marconi Center CA), D Golomb (Neural Networks for Physicists, Minneapolis MN). Wilfrid Rall's development of the theoretical framework of dendritic function was honored by the dedication to Rall of the inaugural issue, Journal of Computational Neuroscience, and by the publication by MIT Press of several selected Rall papers with commentaries (eds., I Segev, J Rinzel, G M Shepherd).

MRB staff were involved with organizing various symposia and meetings. A Sherman was program co-chair of the 1994 Gordon Conference on Theoretical Biology and Biomathematics. R Mejia organized a session on Mathematical Physiology at the First World Congress on Computational Medicine, Public Health and Biotechnology, Austin, TX, April, 1994. J Rinzel served on the program committee for the Computation and Neural Systems meetings: CNS*93, Washington DC; CNS*94, Monterey CA. 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. M Rush and P Pinsky received their Ph D degrees (Univ Maryland, College Park, Applied Mathematics) for thesis research carried out at the NIH and directed by J Rinzel.

During FY 94, two senior staff retired: Drs. Josˇ Gonzalez-Fernandez and Wilfrid Rall, after 32 and 37 years in MRB, respectively. Also of significance, the Branch has relocated to the BSA Bldg (corner of Cedar Ln and Wisconsin Ave) after many years in Bldg 31. Design and organization of the new space were carried out by the team of Rall, Rinzel, and Sherman.

Network Neurobiology

Rhythmogenesis in thalamic networks. We continue to develop our model for the genesis of sleep spindle oscillations. Considering the isolated reticular thalamic nucleus (RE), we address through modeling the hypothesis of Steriade and colleagues that it is the generator of spindle oscillations. Each model neuron exhibits post-inhibitory rebound and is coupled to all the other neurons via GABA-A and GABA-B inhibitory synapses. An isolated network of identical RE cells typically exhibits partial synchrony as it segregates itself into clusters; each neuron tends to burst only every two or more cycles of the population voltage. Blocking the GABA-A synapses or increasing their reversal potential can produce a fully synchronized network. Excitation from thalamocortical (TC) cells results in synchronized oscillations that are robust to heterogeneity and noise. Considering a network of RE cells and TC cells, we model an in vitro experiment by von Krosigk et al (Science 261, 1993). The RE cells inhibit RE and TC cells via GABA-A and GABA-B synapses, while the TC cells excite the RE cells with AMPA synapses. In the network, RE cells burst almost every cycle, while TC cells burst at a lower firing rate but in synchrony with the RE cells. The TC firing rate decreases as these neurons are hyperpolarized. Blocking GABA-A inhibition brings the network to an asynchronized state, in which TC cells burst at low frequency. Checking the effect of sparseness on the system dynamics, we find a transition from an asynchronized to highly synchronized state as the number of average input connections to the cells increases. (D Golomb, J Rinzel, and X-J Wang: U Pittsburgh)

Epileptiform activity in hippocampal networks. We continue to explore simplified models of hippocampal neurons and neural networks. Simulation studies of excitatory hippocampal networks with NMDA and AMPA synapses show that synchronized population bursting (as seen in in vitro seizure models) persists in the face of considerable cell heterogeneity. Desynchronization upon removal of AMPA was shown to depend on both the level of heterogeneity and the intrinsic dynamics of the neurons (i.e., periodic versus chaotic). We have developed various measures which quantify the level of synchrony of network solutions. We analyze and compare the analytic properties of these measures and assess their performance when applied to hippocampal networks. We have now added inhibitory cells to our hippocampal network model and are exploring various phenomena including partially synchronized bursting. (P Pinsky and J Rinzel)

Experimentally-based model of sub-threshold oscillations in the inferior olive nucleus. The inferior olivary neurons are involved in motor tasks such as coordination and learning by generating rhythmic activity at the cerebellar level. This rhythmic activity is expressed, in vitro, as a sub-threshold oscillation (STO) of the membrane potential. Indirect evidence suggests that the STOs are due to both the biophysical properties of the neurons and the electrotonic coupling between these neurons. This work has two parts. In the experimental part, the properties of the currents that participate in the generation of the STO were characterized using the voltage clamp technique in the whole cell patch clamp configuration. In the theoretical part, these data are incorporated into a network of electrically coupled neurons. Starting with a model of two neurons, we show that quiescent neurons can generate STOs when they are electrically coupled, provided that they differ in their ion channel densities. In a large network of such neurons, we show that several different mechanisms could underly STO generation as observed in the olivary nucleus. The model suggests experimental ways to distinguish between these possible mechanisms. Furthermore, the model provides important insights regarding the relations between the non-linear dynamics of the single neurons and the emergent behavior of a population of these neurons. (J Rinzel, and Y Manor, I Segev, Y Yarom: Hebrew U, Jerusalem)

The lamprey central pattern generator (CPG) for swimming. The isolated spinal cord of the lamprey can produce periodic rostral-to-caudal waves of neural activity. This activity can be characterized by measuring the times at which bursts of action potentials occur at different motor nerves along the cord. We are developing a time series model which relates the burst times on one wave to burst times on the previous wave. There are two sources of noise in the model: intrinsic noise, which can be attributed to fluctuations in the CPG's rhythm, and output noise. The model contains a matrix describing the interaction between segments. We are presently fitting the model's parameters to experimental data. (T Kiemel, A Cohen: Univ MD and N Mellon: UCLA)

Cellular Neurobiology

Rall's theoretical foundation of dendritic function. Over the past 40 years, Dr. Wilfrid Rall (of MRB, NIDDK since 1957) has developed the theoretical framework for understanding the function of neuronal dendrites. In order to recognize Rall's work, and to provide a valuable reference source for neuroscience researchers, we have prepared a book through MIT Press which reprints a number of key Rall papers and provides retrospective commentary by several different contributors. Rall formulated and analyzed the equations of cable theory for complex branching dendritic trees, developed the idealized model of the "equivalent cylinder", obtained usable formulae for estimating cable parameters from experimental measurements of somatic potential, and pioneered the use of the compartmental method for numerical simulation of spatio-temporal activity throughout the dendrites of a neuron model. Rall's applications of this theory and methodology have led to significant key insights into dendritic function, including: integration of spatio-temporal patterns of synaptic inputs in motoneurons, dendrodendritic interactions in the olfactory bulb, active membrane properties in dendritic spines and their roles in plasticity and learning. (J Rinzel, W Rall, I Segev: Hebrew Univ, G Shepherd: Yale Univ)

Dendritic origin of bistability of motoneuron firing patterns. Under certain conditions, bistable oscillatory behavior has been observed in vertebrate motoneurons, such as cat lumbar and turtle motoneurons. The two stable oscillatory states can differ in frequency by as much as 10Hz. This bistability is present in intact animals and can be induced in reduced preparations by application of pharmacological agents such as serotonin, noradrenaline, or the potassium channel blocker TEA. When action potentials are depressed by TTX, a plateau potential that provides the cell with two stable states is observed. We hypothesize that the bistability observed in the soma of these cells is due to bistable kinetics in the dendrites. In particular, the higher frequency oscillations, or the depolarized plateau under TTX, occur because the dendrites are in a depolarized stable state. To investigate this hypothesis, we constructed an idealized model of a motoneuron that consists of two compartments. The soma is represented by one compartment containing repetitive spiking kinetics. The dendrites are lumped into the second compartment and display two stable steady solutions. The compartments are coupled electrotonically. For nonzero coupling conductance, bistable periodic solutions exist in the soma compartment. We are exploring this idealized model to determine the dependence of the existence of bistable behavior on coupling parameters and kinetics of the compartments and the necessity for spatial differentiation of soma and dendritic properties. We are also developing a more biophysically accurate motoneuron model with the aim of elucidating the specific membrane properties responsible for the bistable behavior. Our effort involves consultative interaction with various experimentalist neuroscientists: O Kiehn: Panum Inst, Copenhagen and P Schwindt: U Washington). (V Booth and J Rinzel)

NMDA-induced bursting in dopamine neurons. Burst firing of dopamine neurons in vitro is induced by the glutamate agonist N-methyl-D aspartate (NMDA). The hyperpolarization between bursts is believed due to Na extrusion by a ouabain-sensitive pump. We formulate and explore a theoretical model for this novel mechanism of burst generation. We show that interaction between the regenerative, inward NMDA-mediated current and the outward Na-pump current is sufficient to generate the slow oscillation (0.5 Hz) underlying the burst. The region of negative slope in the I-V relation of the NMDA channel in the presence of Mg is indispensable for the occurrence of this slow rhythm. We find that at least 2 spatial compartments are required: a soma where action potentials (APs) are produced and a dendrite where the slow rhythm is generated. The time scale of Na- handling in the dendrite determines the burst period. In the absence of NMDA, the model shows tonic spiking (2 Hz) which is insensitive to Na pump inhibitors. When NMDA is present, tonic spiking is replaced by repetitive bursting. Tetrodotoxin blocks the APs but leaves the slow rhythm unchanged. Na pump inhibitors tranform bursting back into tonic firing. When the soma is voltage-clamped, slow oscillations in current, which are generated in the dendritic compartment, are still present. These results are in good agreement with experimental observations. nsights obtained with our model may apply to other neuron types where bursting appears to involve NMDA channel activity (Y-X Li, R Bertram, and J Rinzel)

Oscillatory Activity of Secretory Cells

Acetylcholine effects on insulin secretion. We complement our previous modeling of regulation of insulin secretion by glucose in pancreatic beta-cells with work on modulation by vagal acetylcholine (ACh). Visual and other prandial stimuli use this pathway to enhance insulin secretion prior to a rise in plasma glucose. In vitro measurements of membrane potential in pancreatic islets show depolarization and an increase in burst frequency and a corresponding dramatic increase of intracellular Ca++ (Soria). Based on recent observations (Atwater and Mears; others) that poisoning the endoplasmic reticulum (ER) Ca-ATPase with thapsigargin (Tg) depolarizes islets, we postulate that release of Ca++ from the ER caused by Tg or by ACh-generated IP3, activates an inward Na+ current. This single new hypothesis suffices to explain all the above data. ER emptying and refilling also introduces a new, slow timescale into the dynamics that may account for the characteristic "bi-phasic" transients upon elevation and reduction of glucose, hitherto a mystery. (A Sherman, R Bertram, P Smolen, I Atwater:LCBG, NIDDK, D Mears LCBG, NIDDK, and B Soria:U Alicante, Alicante)

Electrophysiological patch-clamp protocol. We have completed work begun several years ago on voltage-clamping single beta-cells within an intact pancreatic islet. This challenging approach is motivated by the desire to circumvent problems inherent in deducing islet behavior from patch clamp measurements taken from isolated cells at lower temperatures in a less physiological milieu. Although artifactual coupling currents from unclamped neighbors of the impaled cell are introduced, our computer simulations show that this can be overcome by a standard leak-subtraction protocol: the coupling currents behave approximately linearly when the neighbors are in their silent phases. These simulations have also suggested a method for estimating coupling conductance in situ by measuring the magnitude of invading burst currents. Mears is presently applying this method experimentally. (A Sherman, and from U Houston: C Stokes and L Xu, and from LCBG, NIDDK: I Atwater, D Mears, and E Rojas)

Slow Ca++ oscillations in pancreatic beta-cells. We continue to examine slow oscillations (period 2 - 20 min) of Ca++ and insulin secretion in pancreatic beta-cells. These may supplant or coexist with faster oscillations driven by bursting electrical activity (BEA). More than one mechanism may be involved, as one group has reported that the mitochondrial fuel alpha-ketoisocaproate induces oscillations whereas others propose that such oscillations are glycolytic in origin. We have developed and explored a detailed model of oscillating glycolysis based on allosteric interactions of phosphofructokinase with substrate, product, AMP, and ATP. We have also coupled this model to models for BEA through glycolytic modulation of ATP concentration and hence of the ATP-sensitive K+ conductance. This modulates Ca++ influx and causes oscillations of Ca++ in-phase with ATP, in agreement with data of Corkey et al. Uptake and release of Ca++ from the ER are also required to give observed Ca++ time courses. (P Smolen and A Sherman)

Agonist-induced and membrane potential-generated calcium oscillations in pituitary gonadotrophs. Based on the Hodgkin-Huxley (HH)-like formalism of agonist-induced calcium oscillations, we have already developed a model for the calcium (Ca) responses in gonadotrophs when stimulated with increasing doses of hormonal signal. We further analyzed the spontaneous fluctuations in Ca resulting from the plasma membrane (PM) voltage oscillations. Based on patch-clamp experimental data for the ionic currents, we have developed a quantitative model describing the PM electrical activity and its link to the accompanying oscillations in Ca. We conclude that each Ca fluctuation is produced by the Ca-entry during a single action potential (AP). This calcium concentrates mostly within a thin shell-layer neighboring the PM. We show that, due to the influence of Ca-entry on the Ca-activated K channels, the frequency and the temporal profile of the APs are not only determined by the PM channels, but also by the spatial distribution of the intracellular Ca (Y-X Li and J Rinzel). The dynamic coupling of the agonist-induced Ca oscillations in the cytoplasm and the PM potential oscillations are also studied. We show that the PM Ca pump activity is crucial in determining the steady AP firing frequency. The model reproduces experimental observations at different levels of agonist stimulation with striking agreement. It also shows that, in agreement with experiments, the Ca-entry during each AP gives a stronger phase shift of the cytoplasmic oscillator when the content of intracellular Ca pool is reduced. (Y-X Li, J Rinzel, S Stojilkovic:ERRB/NICHD, J Keizer:UC Davis)

Renal Physiology

Role of the kidney in acid/base balance. We have continued development of a multinephron model for acid/base balance in the whole kidney as experiments to measure the necessary parameters have continued. A model of Henle's loop embedded in an interstitium has been used to study transport in a short circuited preparation. Results have been consistent with those obtained with an open model of Henle's loop in a specified bath or interstitium and with a whole kidney model of the concentrating mechanism. A single population acid/base model has been extended to include the cortical region andcollecting duct. Physiologic parameters for the cortical segments are being sought. (R Mejia and M Knepper:NHLBI)

Concentrating mechanism. Oxytocin has been shown to increase water permeability in perfused rat terminal inner medullary collecting duct at low concentration (0.1 nM) and to inhibit at higher concentration (10 nM). We have used a multinephron model with five populations to study the composition of urine as a function of water permeability of the collecting duct. The model shows urine osmolality to be near maximal for 0.1 nM oxytocin, and total solute mass excretion to peak at basal values and decrease as the water permeability increases. The inhibitory effect on water permeability of higher concentrations of oxytocin (and vasopressin, not studied here) is associated with binding to an oxytocin receptor, to a novel vasopressin receptor, or both. (R Mejia, C-L Chou and M Knepper:NHLBI)


Metabolic control of vasomotion during exercise. It has been observed on the terminal arterioles that in going from rest to exercise the off flow decreases, and the on flow period increases. We can reproduce these features by using the following model. During the arteriolar wall relaxation phase the presence of blood flow results in the conversion of ADP to ATP; the ATP eventually results in the closure of the KATP-sensitive smooth muscle cell membrane channels. This will depolarize the cell membrane and promote Ca++ entry, the vessel wall will contract, with the resulting decrement or absence of blood flow. This will, in turn, decrease ATP. Eventually the KATP-sensitive channels will open; the vessel wall will relax, and the blood flow will restart or increase. The process, then, repeats. During exercise the decrease of tissue ATP is faster than during rest, a lower level of ATP will be reached sooner: the flow off phase will shorten. With the reappearance of the blood flow ADP will reconvert to ATP. However, the rate of this reconversion is smaller than for rest: the flow on phase will lengthen. (J Gonzalez-Fernandez)

Classification of bursting

We have developed a topological classification scheme for bursting oscillations by calculating a two-parameter plane of all possible dynamic behaviors of simple (two-variable) fast, spike-generating systems. Combined with one or more appropriate slow variables, these constitute all possible bursting oscillators within a certain family. Each horizontal cut through the plane represents a particular bursting system. We have demonstrated all the cases, including two previously unknown types, with a single simple fast sub-system. We also find that bursters cannot be adequately classified given phenomenological characteristics from experiment, such as the voltage record and phase resetting properties. Although this completes the taxonomy begun by Rinzel, there are examples of interest outside the family considered. (A Sherman, R Bertram, M Butte and T Kiemel: U MD)

Bifurcations of differential equations with fast and slow variables

In studying systems of equations with fast and slow variables (as in models for bursting oscillations), one often first considers the slow variables as parameters (i.e., as fixed) and studies the fast system. If the fast system has some type of bifurcation we are then interested in the nature of the full system bifurcation, if in fact there is one. We have proved analytically certain relations about the bifurcations of the fast system and the bifurcations (or non-bifurcations) of the full system; the type of bifurcation, or the criticality of the bifurcation, will not always be the same for the fast versus full system. (P Pinsky).

Propagation failure

We investigated the phenomenon of propagation failure in spatially discrete reaction-diffusion systems with excitable (FitzHugh-Nagumo) kinetics. This system is an idealized model of myelinated nerve axons or myocardial tissue. In the limit of weak coupling strength, we constructed a traveling pulse solution. The asymptotic solution for the first three cells was studied in detail. A single Riccati equation describes propagation through each cell. From this equation, we obtained a first approximation for the critical coupling strength below which the pulse is blocked. Near this critical coupling strength, propagation or failure results from a slow passage near a limit point leading either to a successful jump (propagation) or to capture by a slowly varying steady state (failure). The three-cell analysis suggested an algorithmic method for analyzing propagation through a chain of many cells. Failure may appear at any cell but is described by the same Riccati equation. Results from the algorithm compared well with full numerical solutions. (V Booth and T Erneux: U Libre de Bruxelles, Brussels)

Cell energetics

We have previously described the effect of changing transport properties on the concentration profiles of high energy phosphate compounds within single cells using a reaction-diffusion model. During the past year, a model with cylindrical geometry has been ported from a large mainframe (AIX/3090) to a workstation environment (RS/6000 and SUN/Sparc). (R Mejia and R Lynch:U Arizona)

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