In this paper, a model of signaling pathways involving G proteins is investigated. The model inco... more In this paper, a model of signaling pathways involving G proteins is investigated. The model incorporates reaction-diffusion mechanisms in which various reactants participate inside and on the extra-cellular surface membrane. The messenger molecules may diffuse over the surface of the cell membrane and signal transduction across the cell membrane is mediated by membrane receptor bound proteins which connect the genetically controlled biochemical intra-cellular reactions to the production of the second messenger, leading to desired functional responses. Dynamic and steady-state properties of the model are then investigated through weakly nonlinear stability analysis. Turing-type patterns are shown to form robustly under different delineating conditions on the system parameters. The theoretical predictions are then discussed in the context of some recently reported experimental evidence.
A plaque is an accumulation and swelling in the artery walls and typically consists of cells, cel... more A plaque is an accumulation and swelling in the artery walls and typically consists of cells, cell debris, lipids, calcium deposits and fibrous connective tissue. A person is likely to have many plaques inside his/her body even if they are healthy. However plaques may become “vulnerable”, “high-risk” or “thrombosis-prone” if the person engages in a high-fat diet and does not exercise regularly. Such plaques are characterized by a large lipid core, a thin fibrous cap and the presence of macrophages [1]. Vulnerable plaques have an increased likelihood of rupturing and causing cardiac infarction (a heart attack). In fact, about 70% of fatal heart attacks are caused by plaque rupture [3]. Hence their study is of great medical importance. It is the goal of our working group to develop models to understand the growth and rupture of vulnerable plaques. The growth of a plaque is a very complicated process, coupling together cell biochemistry [7], solid mechanics [10] and fluid-structure int...
Many classes of reaction-diffusion systems have been shown to have traveling-wave solutions. For ... more Many classes of reaction-diffusion systems have been shown to have traveling-wave solutions. For a class of such systems for which a comparison theorem can be used, we establish a wave stability result which roughly states that these wave solutions are asymptotically stable to perturbations which lie in some weighted L p {L_p} -space if their speeds are sufficiently large. We then apply this result to some excitable systems, namely a model of the Belousov-Zhabotinskii reaction, a substrate-inhibition biochemical model, and a class of models recently studied by Fife.
Using a comparison theorem technique, we study the long time behavior of certain classes of nonli... more Using a comparison theorem technique, we study the long time behavior of certain classes of nonlinear difference-differential systems. Zero is a solution for these systems. We are concerned in this paper with conditions forcing nonconvergence to zero of solutions as time approaches infinity; that is, we obtain threshold properties of the systems. The results parallel results by Aronson and Weinberger on reaction-diffusion equations somewhat, and the study was motivated by consideration of models for myelinated nerve axons. 1. Introduction. In this paper we study the long time behavior of solutions of nonlinear difference-differential systems of the form *
The Journal of the Acoustical Society of America, 1989
A Pacinian corpuscle (PC) is a rapidly adapting mechanoreceptor that resides in the skin and othe... more A Pacinian corpuscle (PC) is a rapidly adapting mechanoreceptor that resides in the skin and other tissues. The transduction process in the PC consists of a mechanical filtering by the capsule that surrounds the dendrite, along with a receptor membrane component. A model is described that accounts for these. The layer structure of the capsule is taken as formed from concentric elastic membranes with the interlamellar spaces filled with a viscous incompressible fluid. From this the hoop strain in the receptor membrane is found and this serves as the input to a receptor model, which has a strain-sensitive activation mechanism for the transducer conductance. The transient behavior of the receptor potential due to various compression stimuli to the capsule is investigated. Of particular interest is how the viscoelastic properties of the capsule contribute to the adaptive behavior of the PC. To this end the dependence of the hoop strain on the rate of compression and the imposed strain a...
Numerous biological interactions, such as interactions between T cell receptors or antibodies wit... more Numerous biological interactions, such as interactions between T cell receptors or antibodies with antigens, interactions between enzymes and substrates, or interactions between predators and prey are often not strictly specific. In such less specific, or "sloppy," systems, referred to here as degenerate systems, a given unit of a diverse resource (antigens, enzymatic substrates, prey) is at risk of being recognized and consumed by multiple consumers (lymphocytes, enzymes, predators). In this study, we model generalized degenerate consumer-resource systems of Lotka-Volterra and Verhulst types. In the degenerate systems of Lotka-Volterra, there is a continuum of types of consumer and resource based on variation of a single trait (characteristic, or preference). The consumers experience competition for a continuum of resource types. This non-local interaction system is modeled with partial differential-integral equations and shows spontaneous selfstructuring of the consumer population that depends on the degree of interaction degeneracy between resource and consumer, but does not mirror the distribution of resource. We also show that the classical Verhulst (i.e. logistic) single population model can be generalized to a degenerate model, which shows qualitative behavior similar to that in the degenerate Lotka-Volterra model. These results provide better insight into the dynamics of selective systems in biology, suggesting that adaptation of degenerate repertoires is not a simple "mirroring" of the environment by the "fittest" elements of population.
Dendrites of nerve cells have membranes with spatially distributed densities of ionic channels an... more Dendrites of nerve cells have membranes with spatially distributed densities of ionic channels and hence non-uniform conductances. These conductances are usually represented as constant parameters in neural models because of the difficulty in experimentally estimating them locally. In this paper we investigate the inverse problem of recovering a single spatially distributed conductance parameter in a one-dimensional diffusion (cable) equation through a new use of a boundary control method. We also outline how our methodology can be extended to cable theory on finite tree graphs. The reconstruction is unique.
A certain pacific fish has a thalamic nerve cell with a unique dendritic geometry. Instead of the... more A certain pacific fish has a thalamic nerve cell with a unique dendritic geometry. Instead of the cell having a branching dendritic structure, the large cell of the corpus glomerulosum, a part of the thalamus considered associated with the fish's vision, has generally a single dendritic stalk with a large bulbous tip. We formulate a cable theory model that incorporates the geometry, and then we solve the problem for a single localized synaptic current source at the bulb. From the solution representation we numerically examine the characteristics of the postsynaptic potential due to the presence of the bulbous tip.
We extend on the work developed by R.R. Poznanski and J. Bell from a linearized somatic persisten... more We extend on the work developed by R.R. Poznanski and J. Bell from a linearized somatic persistent sodium current source to a non-linear representation of the dendritic Na P current source associated with a small number of persistent sodium channels. The main objective is to investigate the modulation in the ampli®cation of excitatory postsynaptic potentials (EPSPs) in dendrites studded with persistent sodium channels. The relation between membrane potential (V) and persistent sodium current density (I NaP) is approximated heuristically with a sigmoidal function and the resultant cable equation is solved analytically using a regular perturbation expansion and GreenÕs function techniques. The transient simulated (nonevoked) response is found as a result of current injection in the form of synaptically induced voltage change located at a distance from the recording site in a cable with a uniform distribution of ion channel densities per unit length of cable (the so-called`hot-spots') and with the conductance of each hot-spot (i.e., number of channels per hot-spot) assumed to be a constant. The results show an ampli®cation in the observed EPSPs to be compatible with the experimentally derived estimates, and in addition a saturation in the ampli®cation is observed indicating an optimum number of ionic channels.
The persistent sodium current density (I NaP) at the soma measured with the`whole-cell' patch-cla... more The persistent sodium current density (I NaP) at the soma measured with the`whole-cell' patch-clamp recording method is linearized about the resting state and used as a current source along the dendritic cable (depicting the spatial distribution of voltage-dependent persistent sodium ionic channels). This procedure allows time-dependent analytical solutions to be obtained for the membrane depolarization. Computer simulated response to a dendritic current injection in the form of synaptically-induced voltage change located at a distance from the recording site in a cable with unequally distributed persistent sodium ion channel densities per unit length of cable (the so-called`hot-spots') is used to obtain conclusions on the density and distribution of persistent sodium ion channels. It is shown that the excitatory postsynaptic potentials (EPSPs) are ampli®ed if hot-spots of persistent sodium ion channels are spatially distributed along the dendritic cable, with the local density of I NaP with respect to the recording site shown to spe-ci®cally increase the peak amplitude of the EPSP for a proximally placed synaptic input, while the spatial distribution of I NaP serves to broaden the time course of the ampli®ed EPSP. However, in the case of a distally positioned synaptic input, both local and nonlocal densities yield an approximately identical enhancement of EPSPs in contradiction to the computer simulations performed by Lipowsky et al. [J. Neurophysiol. 76 (1996) 2181]. The results indicate that persistent sodium channels produce EPSP am-pli®cation even when their distribution is relatively sparse (i.e., approximately 1±2% of the transient sodium channels are found in dendrites of CA1 hippocampal pyramidal neurons). This gives a strong impetus for the use of the theory as a novel approach in the investigation of synaptic integration of signals in active dendrites represented as ionic cables.
Dendritic and axonal processes of nerve cells, along with the soma itself, have membranes with sp... more Dendritic and axonal processes of nerve cells, along with the soma itself, have membranes with spatially distributed densities of ionic channels of various kinds. These ionic channels play a major role in characterizing the types of excitable responses expected of the cell type. These densities are usually represented as constant parameters in neural models because of the difficulty in experimentally estimating them. However, through microelectrode measurements and selective ion staining techniques, it is known that ion channels are non-uniformly spatially distributed. This paper presents a non-optimization approach to recovering a single spatially non-uniform ion density through use of temporal data that can be gotten from recording microelectrode measurements at the ends of a neural fiber segment of interest. The numerical approach is first applied to a linear cable model and a transformed version of the linear model that has closed-form solutions. Then the numerical method is shown to be applicable to non-linear nerve models by showing it can recover the potassium conductance in the Morris-Lecar model for barnacle muscle, and recover the spine density in a continuous dendritic spine model by Baer and Rinzel.
A functional differential equation which is nonlinear and involves forward and backward deviating... more A functional differential equation which is nonlinear and involves forward and backward deviating arguments is solved numerically. The equation models conduction in a myelinated nerve axon in which the myelin completely insulates the membrane, so that the potential change jumps from node to node. The equation is of first order with boundary values given at t = +/- infinity. The problem is approximated via a difference scheme which solves the problem on a finite interval by utilizing an asymptotic representation at the endpoints, cubic interpolation and iterative techniques to approximate the delays, and a continuation method to start the procedure. The procedure is tested on a class of problems which are solvable analytically to access the scheme's accuracy and stability, then applied to the problem that models propagation in a myelinated axon. The solution's dependence on various model parameters of physical interest is studied. This is the first numerical study of myelinated nerve conduction in which the advance and delay terms are treated explicitly.
Dendrites of nerve cells have membranes with spatially distributed densities of ionic channels an... more Dendrites of nerve cells have membranes with spatially distributed densities of ionic channels and hence non-uniform conductances. These conductances are usually represented as constant parameters in neural models because of the difficulty in experimentally estimating them locally. In this paper we investigate the inverse problem of recovering a single spatially distributed conductance parameter in a one-dimensional diffusion (cable) equation through a new use of a boundary control method. We also outline how our methodology can be extended to cable theory on finite tree graphs. The reconstruction is unique.
In this paper, a model of signaling pathways involving G proteins is investigated. The model inco... more In this paper, a model of signaling pathways involving G proteins is investigated. The model incorporates reaction-diffusion mechanisms in which various reactants participate inside and on the extra-cellular surface membrane. The messenger molecules may diffuse over the surface of the cell membrane and signal transduction across the cell membrane is mediated by membrane receptor bound proteins which connect the genetically controlled biochemical intra-cellular reactions to the production of the second messenger, leading to desired functional responses. Dynamic and steady-state properties of the model are then investigated through weakly nonlinear stability analysis. Turing-type patterns are shown to form robustly under different delineating conditions on the system parameters. The theoretical predictions are then discussed in the context of some recently reported experimental evidence.
A plaque is an accumulation and swelling in the artery walls and typically consists of cells, cel... more A plaque is an accumulation and swelling in the artery walls and typically consists of cells, cell debris, lipids, calcium deposits and fibrous connective tissue. A person is likely to have many plaques inside his/her body even if they are healthy. However plaques may become “vulnerable”, “high-risk” or “thrombosis-prone” if the person engages in a high-fat diet and does not exercise regularly. Such plaques are characterized by a large lipid core, a thin fibrous cap and the presence of macrophages [1]. Vulnerable plaques have an increased likelihood of rupturing and causing cardiac infarction (a heart attack). In fact, about 70% of fatal heart attacks are caused by plaque rupture [3]. Hence their study is of great medical importance. It is the goal of our working group to develop models to understand the growth and rupture of vulnerable plaques. The growth of a plaque is a very complicated process, coupling together cell biochemistry [7], solid mechanics [10] and fluid-structure int...
Many classes of reaction-diffusion systems have been shown to have traveling-wave solutions. For ... more Many classes of reaction-diffusion systems have been shown to have traveling-wave solutions. For a class of such systems for which a comparison theorem can be used, we establish a wave stability result which roughly states that these wave solutions are asymptotically stable to perturbations which lie in some weighted L p {L_p} -space if their speeds are sufficiently large. We then apply this result to some excitable systems, namely a model of the Belousov-Zhabotinskii reaction, a substrate-inhibition biochemical model, and a class of models recently studied by Fife.
Using a comparison theorem technique, we study the long time behavior of certain classes of nonli... more Using a comparison theorem technique, we study the long time behavior of certain classes of nonlinear difference-differential systems. Zero is a solution for these systems. We are concerned in this paper with conditions forcing nonconvergence to zero of solutions as time approaches infinity; that is, we obtain threshold properties of the systems. The results parallel results by Aronson and Weinberger on reaction-diffusion equations somewhat, and the study was motivated by consideration of models for myelinated nerve axons. 1. Introduction. In this paper we study the long time behavior of solutions of nonlinear difference-differential systems of the form *
The Journal of the Acoustical Society of America, 1989
A Pacinian corpuscle (PC) is a rapidly adapting mechanoreceptor that resides in the skin and othe... more A Pacinian corpuscle (PC) is a rapidly adapting mechanoreceptor that resides in the skin and other tissues. The transduction process in the PC consists of a mechanical filtering by the capsule that surrounds the dendrite, along with a receptor membrane component. A model is described that accounts for these. The layer structure of the capsule is taken as formed from concentric elastic membranes with the interlamellar spaces filled with a viscous incompressible fluid. From this the hoop strain in the receptor membrane is found and this serves as the input to a receptor model, which has a strain-sensitive activation mechanism for the transducer conductance. The transient behavior of the receptor potential due to various compression stimuli to the capsule is investigated. Of particular interest is how the viscoelastic properties of the capsule contribute to the adaptive behavior of the PC. To this end the dependence of the hoop strain on the rate of compression and the imposed strain a...
Numerous biological interactions, such as interactions between T cell receptors or antibodies wit... more Numerous biological interactions, such as interactions between T cell receptors or antibodies with antigens, interactions between enzymes and substrates, or interactions between predators and prey are often not strictly specific. In such less specific, or "sloppy," systems, referred to here as degenerate systems, a given unit of a diverse resource (antigens, enzymatic substrates, prey) is at risk of being recognized and consumed by multiple consumers (lymphocytes, enzymes, predators). In this study, we model generalized degenerate consumer-resource systems of Lotka-Volterra and Verhulst types. In the degenerate systems of Lotka-Volterra, there is a continuum of types of consumer and resource based on variation of a single trait (characteristic, or preference). The consumers experience competition for a continuum of resource types. This non-local interaction system is modeled with partial differential-integral equations and shows spontaneous selfstructuring of the consumer population that depends on the degree of interaction degeneracy between resource and consumer, but does not mirror the distribution of resource. We also show that the classical Verhulst (i.e. logistic) single population model can be generalized to a degenerate model, which shows qualitative behavior similar to that in the degenerate Lotka-Volterra model. These results provide better insight into the dynamics of selective systems in biology, suggesting that adaptation of degenerate repertoires is not a simple "mirroring" of the environment by the "fittest" elements of population.
Dendrites of nerve cells have membranes with spatially distributed densities of ionic channels an... more Dendrites of nerve cells have membranes with spatially distributed densities of ionic channels and hence non-uniform conductances. These conductances are usually represented as constant parameters in neural models because of the difficulty in experimentally estimating them locally. In this paper we investigate the inverse problem of recovering a single spatially distributed conductance parameter in a one-dimensional diffusion (cable) equation through a new use of a boundary control method. We also outline how our methodology can be extended to cable theory on finite tree graphs. The reconstruction is unique.
A certain pacific fish has a thalamic nerve cell with a unique dendritic geometry. Instead of the... more A certain pacific fish has a thalamic nerve cell with a unique dendritic geometry. Instead of the cell having a branching dendritic structure, the large cell of the corpus glomerulosum, a part of the thalamus considered associated with the fish's vision, has generally a single dendritic stalk with a large bulbous tip. We formulate a cable theory model that incorporates the geometry, and then we solve the problem for a single localized synaptic current source at the bulb. From the solution representation we numerically examine the characteristics of the postsynaptic potential due to the presence of the bulbous tip.
We extend on the work developed by R.R. Poznanski and J. Bell from a linearized somatic persisten... more We extend on the work developed by R.R. Poznanski and J. Bell from a linearized somatic persistent sodium current source to a non-linear representation of the dendritic Na P current source associated with a small number of persistent sodium channels. The main objective is to investigate the modulation in the ampli®cation of excitatory postsynaptic potentials (EPSPs) in dendrites studded with persistent sodium channels. The relation between membrane potential (V) and persistent sodium current density (I NaP) is approximated heuristically with a sigmoidal function and the resultant cable equation is solved analytically using a regular perturbation expansion and GreenÕs function techniques. The transient simulated (nonevoked) response is found as a result of current injection in the form of synaptically induced voltage change located at a distance from the recording site in a cable with a uniform distribution of ion channel densities per unit length of cable (the so-called`hot-spots') and with the conductance of each hot-spot (i.e., number of channels per hot-spot) assumed to be a constant. The results show an ampli®cation in the observed EPSPs to be compatible with the experimentally derived estimates, and in addition a saturation in the ampli®cation is observed indicating an optimum number of ionic channels.
The persistent sodium current density (I NaP) at the soma measured with the`whole-cell' patch-cla... more The persistent sodium current density (I NaP) at the soma measured with the`whole-cell' patch-clamp recording method is linearized about the resting state and used as a current source along the dendritic cable (depicting the spatial distribution of voltage-dependent persistent sodium ionic channels). This procedure allows time-dependent analytical solutions to be obtained for the membrane depolarization. Computer simulated response to a dendritic current injection in the form of synaptically-induced voltage change located at a distance from the recording site in a cable with unequally distributed persistent sodium ion channel densities per unit length of cable (the so-called`hot-spots') is used to obtain conclusions on the density and distribution of persistent sodium ion channels. It is shown that the excitatory postsynaptic potentials (EPSPs) are ampli®ed if hot-spots of persistent sodium ion channels are spatially distributed along the dendritic cable, with the local density of I NaP with respect to the recording site shown to spe-ci®cally increase the peak amplitude of the EPSP for a proximally placed synaptic input, while the spatial distribution of I NaP serves to broaden the time course of the ampli®ed EPSP. However, in the case of a distally positioned synaptic input, both local and nonlocal densities yield an approximately identical enhancement of EPSPs in contradiction to the computer simulations performed by Lipowsky et al. [J. Neurophysiol. 76 (1996) 2181]. The results indicate that persistent sodium channels produce EPSP am-pli®cation even when their distribution is relatively sparse (i.e., approximately 1±2% of the transient sodium channels are found in dendrites of CA1 hippocampal pyramidal neurons). This gives a strong impetus for the use of the theory as a novel approach in the investigation of synaptic integration of signals in active dendrites represented as ionic cables.
Dendritic and axonal processes of nerve cells, along with the soma itself, have membranes with sp... more Dendritic and axonal processes of nerve cells, along with the soma itself, have membranes with spatially distributed densities of ionic channels of various kinds. These ionic channels play a major role in characterizing the types of excitable responses expected of the cell type. These densities are usually represented as constant parameters in neural models because of the difficulty in experimentally estimating them. However, through microelectrode measurements and selective ion staining techniques, it is known that ion channels are non-uniformly spatially distributed. This paper presents a non-optimization approach to recovering a single spatially non-uniform ion density through use of temporal data that can be gotten from recording microelectrode measurements at the ends of a neural fiber segment of interest. The numerical approach is first applied to a linear cable model and a transformed version of the linear model that has closed-form solutions. Then the numerical method is shown to be applicable to non-linear nerve models by showing it can recover the potassium conductance in the Morris-Lecar model for barnacle muscle, and recover the spine density in a continuous dendritic spine model by Baer and Rinzel.
A functional differential equation which is nonlinear and involves forward and backward deviating... more A functional differential equation which is nonlinear and involves forward and backward deviating arguments is solved numerically. The equation models conduction in a myelinated nerve axon in which the myelin completely insulates the membrane, so that the potential change jumps from node to node. The equation is of first order with boundary values given at t = +/- infinity. The problem is approximated via a difference scheme which solves the problem on a finite interval by utilizing an asymptotic representation at the endpoints, cubic interpolation and iterative techniques to approximate the delays, and a continuation method to start the procedure. The procedure is tested on a class of problems which are solvable analytically to access the scheme's accuracy and stability, then applied to the problem that models propagation in a myelinated axon. The solution's dependence on various model parameters of physical interest is studied. This is the first numerical study of myelinated nerve conduction in which the advance and delay terms are treated explicitly.
Dendrites of nerve cells have membranes with spatially distributed densities of ionic channels an... more Dendrites of nerve cells have membranes with spatially distributed densities of ionic channels and hence non-uniform conductances. These conductances are usually represented as constant parameters in neural models because of the difficulty in experimentally estimating them locally. In this paper we investigate the inverse problem of recovering a single spatially distributed conductance parameter in a one-dimensional diffusion (cable) equation through a new use of a boundary control method. We also outline how our methodology can be extended to cable theory on finite tree graphs. The reconstruction is unique.
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Papers by Jonathan Bell