Emergence of stochastic resonance in a two-compartment hippocampal pyramidal neuron model

Journal of Computational Neuroscience - In vitro studies have shown that hippocampal pyramidal neurons employ a mechanism similar to stochastic resonance (SR) to enhance the detection and...     

Exploring how extracellular electric field modulates neuron activity through dynamical analysis of a two-compartment neuron model

To investigate how extracellular electric field modulates neuron activity, a reduced two-compartment neuron model in the presence of electric field is introduced in this study. Depending on neuronal geometric and internal coupling parameters, the behaviors of the model have been studied extensively. The neuron model can exist in quiescent state or repetitive spiking state in response to electric field stimulus. Negative electric field mainly acts as inhibitory stimulus to the neuron, positive weak electric field could modulate spiking frequency and spike timing when the neuron is already active, and positive electric fields with sufficient intensity could directly trigger neuronal spiking in the absence of other stimulations. By bifurcation analysis, it is observed that there is saddle-node on invariant circle bifurcation, supercritical Hopf bifurcation and subcritical Hopf bifurcation appearing in the obtained two parameter bifurcation diagrams. The bifurcation structures and electric field thresholds for triggering neuron firing are determined by neuronal geometric and coupling parameters. The model predicts that the neurons with a nonsymmetric morphology between soma and dendrite, are more sensitive to electric field stimulus than those with the spherical structure. These findings suggest that neuronal geometric features play a crucial role in electric field effects on the polarization of neuronal compartments. Moreover, by determining the electric field threshold of our biophysical model, we could accurately distinguish between suprathreshold and subthreshold electric fields. Our study highlights the effects of extracellular electric field on neuronal activity from the biophysical modeling point of view. These insights into the dynamical mechanism of electric field may contribute to the investigation and development of electromagnetic therapies, and the model in our study could be further extended to a neuronal network in which the effects of electric fields on network activity may be investigated.     

Frequency switching in a two-compartmental model of the dopaminergic neuron

Mid-brain dopaminergic (DA) neurons display two functionally distinct modes of electrical activity: low- and high-frequency firing. The high-frequency firing is linked to important behavioral events in vivo. However, it cannot be elicited by standard manipulations in vitro. We had suggested a two-compartmental model of the DA cell that united data on firing frequencies under different experimental conditions. We now analyze dynamics of this model. The analysis was possible due to introduction of timescale separation among variables. We formulate the requirements for low and high frequencies. We found that the modulation of the SK current gating controls the frequency rise under applied depolarization. This provides a new mechanism that limits the frequency in the control conditions and allows high-frequency responses to depolarization if the SK current gating is downregulated. The mechanism is based on changing Ca^2 + balance and can also be achieved by direct modulation of the balance. Interestingly, such changes do not affect the high-frequency oscillations under NMDA. Therefore, altering Ca^2 + balance allows combining the high-frequency response to NMDA activation with the inability of other treatments to effectively elevate the frequency. We conclude that manipulations affecting Ca^2 + balance are most effective in controlling the frequency range. This modeling prediction gives a clue to the mechanism of the high-frequency firing in the DA neuron in vivo and in vitro.     

A stochstic model for a two-compartment reversible system

This paper discusses a general stochastic model for a two-compartment reversible system with non-homogeneous Poisson inputs, arbitrary residence times at each of the compartments and time-dependent transition probabilities. The probability distributions of the number of particles in each compartment and in the system are obtained together with the number of particles which depart from the system. In addition, various covariance functions with a time lag are obtained. Some of the above obtained results are deduced for time-independent arrivals, exponential residence times and time-independent transition probabilities. Fluctuations of the particles present in the system are also analysed. Similar analysis is provided for the model into which some particles are initially introduced at the system. Some possible applications are discussed at the end.     

A model of a thalamic neuron

We modify our recent three equilibrium-point model of neuronal bursting by a means of a small deformation of the nullclines in the x-y phase plane to give a model that can have as many as five equilibrium points. In this model the middle stable equilibrium point (e.p.) is separated from the outer stable and unstable e.ps by two saddle points. If the system is started at rest at the middle stable e.p. it has the following complex properties: A short suprathreshold current pulse switches the model from a silent state to a bursting state, or to give a single burst, depending on the choice of parameters. A subthreshold depolarizing current step gives a passive response at rest, but if the model is either constantly hyperpolarized or constantly depolarized, then the same current step gives different active responses. At a hyperpolarized level this consists of a burst response that shows refractoriness. At a depolarized level it consists of tonic firing with a linear frequency--current relationship. Hyperpolarization from rest is followed by post-inhibitory rebound. The model responds in a unique and characteristic way to an applied current ramp. These properties are very similar to those that have been recently recorded intracellularly from neurons in the mammalian thalamus. In the x-y phase plane our models of the repetitively firing neuron, the bursting neuron and the thalamic neuron form a progression of models in which the y nullcline in the subthreshold region is deformed once to give the burst neuron model, and a second time to give the thalamic neuron model. Each deformation can be interpreted as corresponding to the inclusion of a slow inward current in the model. As these currents are included so the associated firing properties increase in complexity.     

How to make a mesodiencephalic dopaminergic neuron

Dopaminergic neurons located in the ventral mesodiencephalon are essential for the control of voluntary movement and the regulation of emotion, and are severely affected in neurodegenerative diseases such as Parkinson's disease. Recent advances in molecular biology and mouse genetics have helped to unravel the mechanisms involved in the development of mesodiencephalic dopaminergic (mdDA) neurons, including their specification, migration and differentiation, as well as the processes that govern axonal pathfinding and their specific patterns of connectivity and maintenance. Here, we follow the developmental path of these neurons with the goal of generating a molecular code that could be exploited in cell-replacement strategies to treat diseases such as Parkinson's disease.     

Dendritic synchrony and transient dynamics in a coupled oscillator model of the dopaminergic neuron

Transient increases in spontaneous firing rate of mesencephalic dopaminergic neurons have been suggested to act as a reward prediction error signal. A mechanism previously proposed involves subthreshold calcium-dependent oscillations in all parts of the neuron. In that mechanism, the natural frequency of oscillation varies with diameter of cell processes, so there is a wide variation of natural frequencies on the cell, but strong voltage coupling enforces a single frequency of oscillation under resting conditions. In previous work, mathematical analysis of a simpler system of oscillators showed that the chain of oscillators could produce transient dynamics in which the frequency of the coupled system increased temporarily, as seen in a biophysical model of the dopaminergic neuron. The transient dynamics was shown to be consequence of a slow drift along an invariant subset of phase space, with rate of drift given by a Lyapunov function. In this paper, we show that the same mathematical structure exists for the full biophysical model, giving physiological meaning to the slow drift and the Lyapunov function, which is shown to describe differences in intracellular calcium concentration in different parts of the cell. The duration of transients was long, being comparable to the time constant of calcium disposition. These results indicate that brief changes in input to the dopaminergic neuron can produce long lasting firing rate transients whose form is determined by intrinsic cell properties.     

A two-compartment model for zinc in humans.

The oral absorption of zinc, from a test meal of minced beef, mashed potatoes and peas, have been measured in 19 healthy adults using the radiotracer 65Zn. The oral absorption, expressed as a percentage of the administered dose, was 20 +/- 5% (mean +/- 1 SD) in good agreement with previous results. In a subset of 9 subjects, tracer retention in whole body and whole blood was followed out to one year. The data were fitted to a simple two compartment model yielding total body zinc (TBZn), the zinc content in each of the 2 compartments and zinc turnover. The TBZn values ranged from 15.5 to 35.9 mmol while zinc turnover ranged from 0.043 to 0.073 mmol/d in keeping with results reported for significantly more complicated compartmental models applied to more comprehensive 65Zn tracer data sets. Additionally, TBZn correlated well with total body potassium, a measure of lean body mass, measured by whole body counting of the naturally-occurring potassium radioisotope, 40K. The zinc content of the more rapidly turning over compartment ranged from 3.2 to 5.6 mmol in reasonable agreement with exchangeable zinc pool estimations reported for short term studies using stable zinc isotopes. Therefore, the simple dataset and model employed in the present study yielded information on the short- and long-term behaviour of zinc compatible with both more complex radiotracer studies and analytically more demanding stable isotope studies.     

The effect of variations in sodium conductances on pacemaking in a dopaminergic retinal neuron model

Dopaminergic neurons in the retina show spontaneous tetrodotoxin-sensitive pacemaking, which has been explained by a reduced Hodgkin-Huxley-type computer model. The present study used this model to investigate the effect of variations in transient and persistent sodium conductance values on pacemaking, under variable leakage conductance levels. This study indicated that transient sodium conductance plays an indispensable role in pacemaking, which occurs under conditions in which only a persistent sodium conductance is considerably reduced, thus contributing to a detailed understanding of the relationship between sodium conductance and pacemaking.     

DYRK1A promotes dopaminergic neuron survival in the developing brain and in a mouse model of Parkinson's disease

In the brain, programmed cell death (PCD) serves to adjust the numbers of the different types of neurons during development, and its pathological reactivation in the adult leads to neurodegeneration. Dual-specificity tyrosine-(Y)-phosphorylation regulated kinase 1A (DYRK1A) is a pleiotropic kinase involved in neural proliferation and cell death, and its role during brain growth is evolutionarily conserved. Human DYRK1A lies in the Down syndrome critical region on chromosome 21, and heterozygous mutations in the gene cause microcephaly and neurological dysfunction. The mouse model for DYRK1A haploinsufficiency (the Dyrk1a^+/− mouse) presents neuronal deficits in specific regions of the adult brain, including the substantia nigra (SN), although the mechanisms underlying these pathogenic effects remain unclear. Here we study the effect of DYRK1A copy number variation on dopaminergic cell homeostasis. We show that mesencephalic DA (mDA) neurons are generated in the embryo at normal rates in the Dyrk1a haploinsufficient model and in a model (the mBACtgDyrk1a mouse) that carries three copies of Dyrk1a. We also show that the number of mDA cells diminishes in postnatal Dyrk1a^+/− mice and increases in mBACtgDyrk1a mice due to an abnormal activity of the mitochondrial caspase9 (Casp9)-dependent apoptotic pathway during the main wave of PCD that affects these neurons. In addition, we show that the cell death induced by 1-methyl-4-phenyl-1,2,3,6 tetrahydropyridine (MPTP), a toxin that activates Casp9-dependent apoptosis in mDA neurons, is attenuated in adult mBACtgDyrk1a mice, leading to an increased survival of SN DA neurons 21 days after MPTP intoxication. Finally, we present data indicating that Dyrk1a phosphorylation of Casp9 at the Thr125 residue is the mechanism by which this kinase hinders both physiological and pathological PCD in mDA neurons. These data provide new insight into the mechanisms that control cell death in brain DA neurons and they show that deregulation of developmental apoptosis may contribute to the phenotype of patients with imbalanced DYRK1A gene dosage.The total number of neurons in the brain, and ultimately the size of this organ, depends both on the number of cells that are produced during neurogenesis and the number of neurons that die due to physiological programmed cell death (PCD). Dual-specificity tyrosine-(Y)-phosphorylation regulated kinase 1A (DYRK1A) regulates brain growth in a dose-dependent manner,¹ and indeed, loss-of-function mutations in DYRK1A (minibrain in Drosophila melanogaster) cause microcephaly and several neurological alterations in humans,^{2, 3, 4, 5} mice⁶ and flies.⁷ Accordingly, it has been proposed that haploinsufficiency of DYRK1A is the cause of the microcephaly and developmental delay associated to partial monosomy of chromosome 21 involving DYRK1A.⁸ Moreover, triplication of the gene has been associated to the developmental brain dysfunctions and age-associated neurodegeneration observed in Down syndrome.^{9, 10, 11}Anatomical analysis of adult Dyrk1a mutant mice that model human diseases involving an imbalance in DYRK1A gene dosage (the Dyrk1a^+/− mouse and the mBACtgDyrk1a mouse, carrying one or three functional copies of Dyrk1a, respectively) revealed a positive correlation between Dyrk1a gene copy number, the overall size of the brain and the number of neurons in specific regions.¹ DYRK1A regulates several fundamental neurodevelopmental processes, including proliferation, neuron differentiation and PCD.¹² Overexpression of DYRK1A in neural precursors attenuates proliferation and promotes the differentiation of neurons in different model systems.^{13, 14, 15} Conversely, treatment of neural progenitors with DYRK1A kinase inhibitors increases proliferation.¹⁵ Although these data are consistent with some of the defects in cellularity identified in specific brain regions of Dyrk1a gene copy number mutants, they cannot explain the severe microcephaly evident in mice and humans carrying one functional copy of DYRK1A, or the overall macrocephaly in the mBACtgDyrk1a model carrying three Dyrk1a alleles.^{1, 5} Thus, deregulation of other DYRK1A functions might also contribute to the defects in brain cellularity in these Dyrk1a gene copy number mutants, such as those described in retinal neurons that restrain developmental PCD.¹⁶Dopaminergic (DA) neurons in the substantia nigra (SN) and ventral tegmental area (VTA) have an important role in controlling fine motor actions, as well as in motivation and reward behaviours, and their loss is associated with Parkinson''s disease.¹⁷ In aged Dyrk1a^+/− mice the SN is smaller and contains fewer DA neurons than in wild-type mice.¹⁸ These mutant animals are hypoactive, with altered gait dynamics, and as these defects are evident preweaning and in young animals,^{6, 18, 19} as well as in children with heterozygous mutations in DYRK1A,^{3, 4, 5} they might arise during development.To provide insight into the aetiology of the neurological phenotype caused by DYRK1A haploinsufficiency, here we studied the development of mesencephalic DA (mDA) neurons in Dyrk1a^+/− and mBACtgDyrk1a mouse models. The results obtained show that Dyrk1a copy number variation does not affect the generation of DA neurons, but rather it modifies the number of these neurons that undergo physiological PCD due to an inhibitory effect of the Dyrk1a kinase on the apoptotic activity of caspase9 (Casp9), the initiator caspase in the mitochondrial-dependent apoptotic pathway.²⁰ Thus, deregulation of Casp9-dependent PCD during development may contribute to the brain size defects observed in aneuploidies involving DYRK1A.As inappropriate re-activation of the mitochondrial-dependent apoptotic pathway in mature mDA neurons contributes to the neurodegeneration associated with Parkinson''s disease,²¹ we used the mBACtgDyrk1a mouse model to assess whether basal Dyrk1a-dependent inhibition of Casp9 apoptotic activity could restrain the neurodegeneration induced in vivo by the parkinsonian neurotoxin 1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine (MPTP). Our results show that the apoptotic response to the toxin in mBACtgDyrk1a mice is significantly attenuated, leading to an increase in the number of SN pars compacta DA neurons that resist the pathological insult.     

A two-compartment model of VEGF distribution in the mouse

Vascular endothelial growth factor (VEGF) is a key regulator of angiogenesis--the growth of new microvessels from existing microvasculature. Angiogenesis is a complex process involving numerous molecular species, and to better understand it, a systems biology approach is necessary. In vivo preclinical experiments in the area of angiogenesis are typically performed in mouse models; this includes drug development targeting VEGF. Thus, to quantitatively interpret such experimental results, a computational model of VEGF distribution in the mouse can be beneficial. In this paper, we present an in silico model of VEGF distribution in mice, determine model parameters from existing experimental data, conduct sensitivity analysis, and test the validity of the model. The multiscale model is comprised of two compartments: blood and tissue. The model accounts for interactions between two major VEGF isoforms (VEGF(120) and VEGF(164)) and their endothelial cell receptors VEGFR-1, VEGFR-2, and co-receptor neuropilin-1. Neuropilin-1 is also expressed on the surface of parenchymal cells. The model includes transcapillary macromolecular permeability, lymphatic transport, and macromolecular plasma clearance. Simulations predict that the concentration of unbound VEGF in the tissue is approximately 50-fold greater than in the blood. These concentrations are highly dependent on the VEGF secretion rate. Parameter estimation was performed to fit the simulation results to available experimental data, and permitted the estimation of VEGF secretion rate in healthy tissue, which is difficult to measure experimentally. The model can provide quantitative interpretation of preclinical animal data and may be used in conjunction with experimental studies in the development of pro- and anti-angiogenic agents. The model approximates the normal tissue as skeletal muscle and includes endothelial cells to represent the vasculature. As the VEGF system becomes better characterized in other tissues and cell types, the model can be expanded to include additional compartments and vascular elements.     

Bifurcation analysis of a two-compartment hippocampal pyramidal cell model

The Pinsky-Rinzel model is a non-smooth 2-compartmental CA3 pyramidal cell model that has been used widely within the field of neuroscience. Here we propose a modified (smooth) system that captures the qualitative behaviour of the original model, while allowing the use of available, numerical continuation methods to perform full-system bifurcation and fast-slow analysis. We study the bifurcation structure of the full system as a function of the applied current and the maximal calcium conductance. We identify the bifurcations that shape the transitions between resting, bursting and spiking behaviours, and which lead to the disappearance of bursting when the calcium conductance is reduced. Insights gained from this analysis, are then used to firstly illustrate how the irregular spiking activity found between bursting and stable spiking states, can be influenced by phase differences in the calcium and dendritic voltage, which lead to corresponding changes in the calcium-sensitive potassium current. Furthermore, we use fast-slow analysis to investigate the mechanisms of bursting and show that bursting in the model is dependent on the intermediately slow variable, calcium, while the other slow variable, the activation gate of the afterhyperpolarisation current, does not contribute to setting the intraburst dynamics but participates in setting the interburst interval. Finally, we discuss how some of the described bifurcations affect spiking behaviour, during sharp-wave ripples, in a larger network of Pinsky-Rinzel cells.     

A two-compartment model of osmotic lysis in Plasmodium falciparum-infected erythrocytes

We recently identified a voltage-dependent anion channel on the surface of human red blood cells (RBCs) infected with the malaria parasite, Plasmodium falciparum. This channel, the plasmodial erythrocyte surface anion channel (PESAC), likely accounts for the increased permeability of infected RBCs to various small solutes, as assessed quantitatively with radioisotope flux and patch-clamp studies. Whereas this increased permeability has also been studied by following osmotic lysis of infected cells in permeant solutes, these experiments have been limited to qualitative comparisons of lysis rates. To permit more quantitative examination of lysis rates, we have developed a mathematical model for osmotic fragility of infected cells based on diffusional uptake via PESAC and the two-compartment geometry of infected RBCs. This model, combined with a simple light scattering assay designed to track osmotic lysis precisely, produced permeability coefficients that match both previous isotope flux and patch-clamp estimates. Our model and light scattering assay also revealed Michaelian kinetics for inhibition of PESAC by furosemide, suggesting a 1:1 stoichiometry for their interaction.     

A time-dependent phenomenological model for cell mechano-sensing

Adherent cells normally apply forces as a generic means of sensing and responding to the mechanical nature of their surrounding environment. How these forces vary as a function of the extracellular rigidity is critical to understanding the regulatory functions that drive important phenomena such as wound healing or muscle contraction. In recognition of this fact, experiments have been conducted to understand cell rigidity-sensing properties under known conditions of the extracellular environment, opening new possibilities for modeling this active behavior. In this work, we provide a physics-based constitutive model taking into account the main structural components of the cell to reproduce its most significant contractile properties such as the traction forces exerted as a function of time and the extracellular stiffness. This model shows how the interplay between the time-dependent response of the acto-myosin contractile system and the elastic response of the cell components determines the mechano-sensing behavior of single cells.     

A diffusion model of a neuron and neural nets

In the paper a diffusion model of a neuron is treated. A new, less restrictive than usually, condition of applicability of a diffusion model is presented. As a result the point-process-to-point-process model of a neuron is obtained, which produces an output signal of the same kind as the accepted input signals.     

Impact of protein binding on receptor occupancy: A two-compartment model

In this paper we analyse the impact of protein-, lipid- and receptor-binding on receptor occupancy in a two-compartment system, with proteins in both compartments and lipids and receptors in the peripheral compartment only. We do this for two manners of drug administration: a bolus administration and a constant rate infusion, both into the central compartment. We derive explicit approximations for the time-curves of the different compounds valid for a wide range of realistic values of rate constants and initial concentrations of proteins, lipids, receptors and the drug. These approximations are used to obtain both qualitative and quantitative insight into such critical properties as the distribution of the drug over the two compartments, the maximum receptor occupancy and the area under the drug-receptor complex curve. In particular we focus on assessing the impact of the dissociation constants, K_P, K_L and K_R of the drug with, respectively, the proteins, the lipids and the receptors, the permeability and the surface area of the membrane between compartments, and the rate the drug is eliminated from the system.