Periodicities in the Hodgkin-Huxley model and versions of this model with stochastic input

A field of computational neuroscience develops mathematical models to describe neuronal systems. The aim is to better understand the nervous system. Historically, the integrate-and-fire model, developed by Lapique in 1907, was the first model describing a neuron. In 1952 Hodgkin and Huxley [8] described the so called Hodgkin-Huxley model in the article “A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve”. The Hodgkin-Huxley model is one of the most successful and widely-used biological neuron models. Based on experimental data from the squid giant axon, Hodgkin and Huxley developed their mathematical model as a four-dimensional system of first-order ordinary differential equations. One of these equations characterizes the membrane potential as a process in time, whereas the other three equations depict the opening and closing state of sodium and potassium ion channels. The membrane potential is proportional to the sum of ionic current flowing across the membrane and an externally applied current. For various types of external input the membrane potential behaves differently. This thesis considers the following three types of input: (i) Rinzel and Miller [15] calculated an interval of amplitudes for a constant applied current, where the membrane potential is repetitively spiking; (ii) Aihara, Matsumoto and Ikegaya [1] said that dependent on the amplitude and the frequency of a periodic applied current the membrane potential responds periodically; (iii) Izhikevich [12] stated that brief pulses of positive and negative current with different amplitudes and frequencies can lead to a periodic response of the membrane potential. In chapter 1 the Hodgkin-Huxley model is introduced according to Izhikevich [12]. Besides the definition of the model, several biological and physiological notes are made, and further concepts are described by examples. Moreover, the numerical methods to solve the equations of the Hodgkin-Huxley model are presented which were used for the computer simulations in chapter 2 and chapter 3. In chapter 2 the statements for the three different inputs (i), (ii) and (iii) will be verified, and periodic behavior for the inputs (ii) and (iii) will be investigated. In chapter 3 the inputs are embedded in an Ornstein-Uhlenbeck process to see the influence of noise on the results of chapter 2.

On non-linear dynamics and an optimal control synthesis of the action potential of membranes (ideal and non-ideal cases) of the Hodgkin-Huxley (HH) mathematical model

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

A multi-GPU algorithm for large-scale neuronal networks

Large-scale simulations of parts of the brain using detailed neuronal models to improve our understanding of brain functions are becoming a reality with the usage of supercomputers and large clusters. However, the high acquisition and maintenance cost of these computers, including the physical space, air conditioning, and electrical power, limits the number of simulations of this kind that scientists can perform. Modern commodity graphical cards, based on the CUDA platform, contain graphical processing units (GPUs) composed of hundreds of processors that can simultaneously execute thousands of threads and thus constitute a low-cost solution for many high-performance computing applications. In this work, we present a CUDA algorithm that enables the execution, on multiple GPUs, of simulations of large-scale networks composed of biologically realistic Hodgkin-Huxley neurons. The algorithm represents each neuron as a CUDA thread, which solves the set of coupled differential equations that model each neuron. Communication among neurons located in different GPUs is coordinated by the CPU. We obtained speedups of 40 for the simulation of 200k neurons that received random external input and speedups of 9 for a network with 200k neurons and 20M neuronal connections, in a single computer with two graphic boards with two GPUs each, when compared with a modern quad-core CPU. Copyright (C) 2010 John Wiley & Sons, Ltd.

Clustering through post inhibitory rebound in synaptically coupled neurons

Post inhibitory rebound is a nonlinear phenomenon present in a variety of nerve cells. Following a period of hyper-polarization this effect allows a neuron to fire a spike or packet of spikes before returning to rest. It is an important mechanism underlying central pattern generation for heartbeat, swimming and other motor patterns in many neuronal systems. In this paper we consider how networks of neurons, which do not intrinsically oscillate, may make use of inhibitory synaptic connections to generate large scale coherent rhythms in the form of cluster states. We distinguish between two cases i) where the rebound mechanism is due to anode break excitation and ii) where rebound is due to a slow T-type calcium current. In the former case we use a geometric analysis of a McKean type model to obtain expressions for the number of clusters in terms of the speed and strength of synaptic coupling. Results are found to be in good qualitative agreement with numerical simulations of the more detailed Hodgkin-Huxley model. In the second case we consider a particular firing rate model of a neuron with a slow calcium current that admits to an exact analysis. Once again existence regions for cluster states are explicitly calculated. Both mechanisms are shown to prefer globally synchronous states for slow synapses as long as the strength of coupling is sufficiently large. With a decrease in the duration of synaptic inhibition both systems are found to break into clusters. A major difference between the two mechanisms for cluster generation is that anode break excitation can support clusters with several groups, whilst slow T-type calcium currents predominantly give rise to clusters of just two (anti-synchronous) populations.

On non-linear dynamics and control designs applied to the ideal and non-ideal variants of the Fitzhugh-Nagumo (FN) mathematical model

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Complex chemical dynamics through engineering-like methods

Most of the problems in modern structural design can be described with a set of equation; solutions of these mathematical models can lead the engineer and designer to get info during the design stage. The same holds true for physical-chemistry; this branch of chemistry uses mathematics and physics in order to explain real chemical phenomena. In this work two extremely different chemical processes will be studied; the dynamic of an artificial molecular motor and the generation and propagation of the nervous signals between excitable cells and tissues like neurons and axons. These two processes, in spite of their chemical and physical differences, can be both described successfully by partial differential equations, that are, respectively the Fokker-Planck equation and the Hodgkin and Huxley model. With the aid of an advanced engineering software these two processes have been modeled and simulated in order to extract a lot of physical informations about them and to predict a lot of properties that can be, in future, extremely useful during the design stage of both molecular motors and devices which rely their actions on the nervous communications between active fibres.

SNNAP: A Tool for Teaching Neuroscience

Simulation tools aid in learning neuroscience by providing the student with an interactive environment to carry out simulated experiments and test hypotheses. The field of neuroscience is well suited for the use of simulation tools since nerve cell signaling can be described by mathematical equations and solved by computer. Neural signaling entails the propagation of electrical current along nerve membrane and transmission to neighboring neurons through synaptic connections. Action potentials and synaptic transmission can be simulated and results displayed for visualization and analysis. The neurosimulator SNNAP (Simulator for Neural Networks and Action Potentials) is a simulation environment that provides users with editors for model building, simulator engine and visual display editor. This paper presents several modeling examples that illustrate some of the capabilities and features of SNNAP. First, the Hodgkin-Huxley (HH) model is presented and the threshold phenomenon is illustrated. Second, small neural networks are described with HH models using various synaptic connections available with SNNAP. Synaptic connections may be modulated through facilitation or depression with SNNAP. A study of vesicle pool dynamics is presented using an AMPA receptor model. Finally, a central pattern generator model of the Aplysia feeding circuit is illustrated as an example of a complex network that may be studied with SNNAP. Simulation code is provided for each case study described and tasks are suggested for further investigation.

Respiratory rhythm generation: A modeling study of the respiratory central pattern generator and the phrenic motor neuron

The respiratory central pattern generator is a collection of medullary neurons that generates the rhythm of respiration. The respiratory central pattern generator feeds phrenic motor neurons, which, in turn, drive the main muscle of respiration, the diaphragm. The purpose of this thesis is to understand the neural control of respiration through mathematical models of the respiratory central pattern generator and phrenic motor neurons. ^ We first designed and validated a Hodgkin-Huxley type model that mimics the behavior of phrenic motor neurons under a wide range of electrical and pharmacological perturbations. This model was constrained physiological data from the literature. Next, we designed and validated a model of the respiratory central pattern generator by connecting four Hodgkin-Huxley type models of medullary respiratory neurons in a mutually inhibitory network. This network was in turn driven by a simple model of an endogenously bursting neuron, which acted as the pacemaker for the respiratory central pattern generator. Finally, the respiratory central pattern generator and phrenic motor neuron models were connected and their interactions studied. ^ Our study of the models has provided a number of insights into the behavior of the respiratory central pattern generator and phrenic motor neurons. These include the suggestion of a role for the T-type and N-type calcium channels during single spikes and repetitive firing in phrenic motor neurons, as well as a better understanding of network properties underlying respiratory rhythm generation. We also utilized an existing model of lung mechanics to study the interactions between the respiratory central pattern generator and ventilation. ^

Neurite, a finite difference large scale parallel program for the simulation of the electrical signal propagation in neurites under mechanical loading

With the growing body of research on traumatic brain injury and spinal cord injury, computational neuroscience has recently focused its modeling efforts on neuronal functional deficits following mechanical loading. However, in most of these efforts, cell damage is generally only characterized by purely mechanistic criteria, function of quantities such as stress, strain or their corresponding rates. The modeling of functional deficits in neurites as a consequence of macroscopic mechanical insults has been rarely explored. In particular, a quantitative mechanically based model of electrophysiological impairment in neuronal cells has only very recently been proposed (Jerusalem et al., 2013). In this paper, we present the implementation details of Neurite: the finite difference parallel program used in this reference. Following the application of a macroscopic strain at a given strain rate produced by a mechanical insult, Neurite is able to simulate the resulting neuronal electrical signal propagation, and thus the corresponding functional deficits. The simulation of the coupled mechanical and electrophysiological behaviors requires computational expensive calculations that increase in complexity as the network of the simulated cells grows. The solvers implemented in Neurite-explicit and implicit-were therefore parallelized using graphics processing units in order to reduce the burden of the simulation costs of large scale scenarios. Cable Theory and Hodgkin-Huxley models were implemented to account for the electrophysiological passive and active regions of a neurite, respectively, whereas a coupled mechanical model accounting for the neurite mechanical behavior within its surrounding medium was adopted as a link between lectrophysiology and mechanics (Jerusalem et al., 2013). This paper provides the details of the parallel implementation of Neurite, along with three different application examples: a long myelinated axon, a segmented dendritic tree, and a damaged axon. The capabilities of the program to deal with large scale scenarios, segmented neuronal structures, and functional deficits under mechanical loading are specifically highlighted.

Spatio-temporal filtering properties of a dendritic cable with active spines: a modeling study in the spike-diffuse-spike framework

The spike-diffuse-spike (SDS) model describes a passive dendritic tree with active dendritic spines. Spine-head dynamics is modeled with a simple integrate-and-fire process, whilst communication between spines is mediated by the cable equation. In this paper we develop a computational framework that allows the study of multiple spiking events in a network of such spines embedded on a simple one-dimensional cable. In the first instance this system is shown to support saltatory waves with the same qualitative features as those observed in a model with Hodgkin-Huxley kinetics in the spine-head. Moreover, there is excellent agreement with the analytically calculated speed for a solitary saltatory pulse. Upon driving the system with time varying external input we find that the distribution of spines can play a crucial role in determining spatio-temporal filtering properties. In particular, the SDS model in response to periodic pulse train shows a positive correlation between spine density and low-pass temporal filtering that is consistent with the experimental results of Rose and Fortune [1999, Mechanisms for generating temporal filters in the electrosensory system. The Journal of Experimental Biology 202, 1281-1289]. Further, we demonstrate the robustness of observed wave properties to natural sources of noise that arise both in the cable and the spine-head, and highlight the possibility of purely noise induced waves and coherent oscillations.

Estudo das vias de acesso aos cuidados psiquiátricos na equipa de Sintra

Resumo: O estudo das vias de acesso à consulta de Psiquiatria permite identificar os parceiros mais importantes no acesso dos utentes aos serviços psiquiátricos. O modelo de Goldberg-Huxley considera que o acesso às consultas de Psiquiatria se faz principalmente através dos cuidados de saúde primários. Material e Métodos: Para estudar as vias de acesso aos cuidados psiquiátricos utilizamos a Encounter Form, questionário desenvolvido por Gater. Foi também avaliada a classe social dos utentes utilizando a Escala de Graffar. Este inquérito foi passado na Consulta de Psiquiatria de Sintra a utentes de primeira consulta. A amostra estudada foi de 93 utentes. O objectivo do estudo foi conhecer a trajectória do utente desde que teve necessidade de ser consultado até chegar à consulta de Psiquiatria, os sintomas que determinaram a decisão de procurar ajuda e a influência da classe social no tempo de percurso. Resultados: Observa-se que os utentes passam pela Medicina Geral e Familiar em 71 % dos casos, pela Urgência Psiquiátrica em 16,1 % dos casos, pela Medicina Especializada Hospitalar em 10,7 % dos casos e pela Urgência Geral em 1,1 % dos casos. Na escala de Graffar a classe social prevalente é a média (Classe III). O tempo de percurso foi maior que em estudo similar realizado em 1991. A classe Social III foi a que teve tempo de percurso maior. Conclusões: O estudo conclui que o acesso a esta consulta de Psiquiatria se faz principalmente através da Medicina Geral e Familiar. O tempo de percurso é maior que o desejável por falta de recursos humanos.------- ABSTRACT: Introduction: The study of the Pathways to Psychiatric Care identifies the most important partners in accessing psychiatric services. The Goldberg- Huxley model believes that access to Psychiatric consultation is done preferably through the primary health care. Material and Methods: This survey included 93 first-time users of the Psychiatric Consultation of Sintra. The aim was to study the trajectory of the user since he had felt a need to be consulted until the consultation of Psychiatry, the symptoms that led to the decision to seek help and influence of social class in time spent in pathways. This study used the Encounter Form, a questionnaire developed by Gater. Social class of users was also assessed using the Scale of Graffar. Results: We observed that users have contact with General Practitionaires in 71% of cases, the Psychiatric Urgency in 16.1% of cases, the Hospital Medical Specialist in 10.7% of cases and the General Urgency in 1,1% of cases. On the Graffar scale middle class (Class III) was the most prevalent. The travel time spend in pathways was reater than that obtained in a similar study carried out in 1991. Social Class III group had a greater time spent on pathways. Conclusions: The study concludes that access to this Psychiatric consultation is principally through general practice. The time spent in pathways is greater than desirable due to lack of resources.

Seeing and believing: three episodes in the history of neurophysiology

In this contribution I look at three episodes in the history of neurophysiology that bring out the complex relationship between seeing and believing. I start with Vesalius in the mid-sixteenth century who writes that he can in no way see any cavity in nerves, even in the optic nerves. He thus questions the age-old theory (dating back to the Alexandrians in the third century BC) but, because of the overarching psychophysiology of his time, does not press his case. This conflict between observation and theory persisted for a quarter of a millennium until finally resolved at the beginning of the nineteenth century by the discoveries of Galvani and Volta. The second case is provided by the early history of retinal synaptology. Schultze in 1866 had represented rod spherules and bipolar dendrites in the outer plexiform layer as being separated by a (synaptic) gap, yet in his written account, because of his theoretical commitments, held them to be continuous. Cajal later, 1892, criticized Schultze for this pusillanimity, but his own figure in La Cellule is by no means clear. It was only with the advent of the electron microscopy in the mid-twentieth century that the true complexity of the junction was revealed and it was shown that both investigators were partially right. My final example comes from the Hodgkin-Huxley biophysics of the 1950s. Their theory of the action potential depended on the existence of unseen ion pores with quite complex biophysical characteristics. These were not seen until the Nobel-Prize-winning X-ray diffraction analyses of the early twenty-first century. Seeing, even at several removes, then confirmed Hodgkin and Huxley’s belief. The relation between seeing and believing is by no means straightforward.

Mathematical Modeling to Investigate Antiarrhythmic Drug Side Effects: Rate-Dependence Role in Ionic Currents and Action Potentials Shape in the O’Hara Model

Sudden cardiac death due to ventricular arrhythmia is one of the leading causes of mortality in the world. In the last decades, it has proven that anti-arrhythmic drugs, which prolong the refractory period by means of prolongation of the cardiac action potential duration (APD), play a good role in preventing of relevant human arrhythmias. However, it has long been observed that the “class III antiarrhythmic effect” diminish at faster heart rates and that this phenomenon represent a big weakness, since it is the precise situation when arrhythmias are most prone to occur. It is well known that mathematical modeling is a useful tool for investigating cardiac cell behavior. In the last 60 years, a multitude of cardiac models has been created; from the pioneering work of Hodgkin and Huxley (1952), who first described the ionic currents of the squid giant axon quantitatively, mathematical modeling has made great strides. The O’Hara model, that I employed in this research work, is one of the modern computational models of ventricular myocyte, a new generation began in 1991 with ventricular cell model by Noble et al. Successful of these models is that you can generate novel predictions, suggest experiments and provide a quantitative understanding of underlying mechanism. Obviously, the drawback is that they remain simple models, they don’t represent the real system. The overall goal of this research is to give an additional tool, through mathematical modeling, to understand the behavior of the main ionic currents involved during the action potential (AP), especially underlining the differences between slower and faster heart rates. In particular to evaluate the rate-dependence role on the action potential duration, to implement a new method for interpreting ionic currents behavior after a perturbation effect and to verify the validity of the work proposed by Antonio Zaza using an injected current as a perturbing effect.

A general hazard model for lifetime data in the presence of cure rate

Historically, the cure rate model has been used for modeling time-to-event data within which a significant proportion of patients are assumed to be cured of illnesses, including breast cancer, non-Hodgkin lymphoma, leukemia, prostate cancer, melanoma, and head and neck cancer. Perhaps the most popular type of cure rate model is the mixture model introduced by Berkson and Gage [1]. In this model, it is assumed that a certain proportion of the patients are cured, in the sense that they do not present the event of interest during a long period of time and can found to be immune to the cause of failure under study. In this paper, we propose a general hazard model which accommodates comprehensive families of cure rate models as particular cases, including the model proposed by Berkson and Gage. The maximum-likelihood-estimation procedure is discussed. A simulation study analyzes the coverage probabilities of the asymptotic confidence intervals for the parameters. A real data set on children exposed to HIV by vertical transmission illustrates the methodology.

Arterial wall dosimetry for non-Hodgkin lymphoma patients treated with radioimmunotherapy.

Tumors in non-Hodgkin lymphoma (NHL) patients are often proximal to the major blood vessels in the abdomen or neck. In external-beam radiotherapy, these tumors present a challenge because imaging resolution prevents the beam from being targeted to the tumor lesion without also irradiating the artery wall. This problem has led to potentially life-threatening delayed toxicity. Because radioimmunotherapy has resulted in long-term survival of NHL patients, we investigated whether the absorbed dose (AD) to the artery wall in radioimmunotherapy of NHL is of potential concern for delayed toxicity. SPECT resolution is not sufficient to enable dosimetric analysis of anatomic features of the thickness of the aortic wall. Therefore, we present a model of aortic wall toxicity based on data from 4 patients treated with (131)I-tositumomab. METHODS: Four NHL patients with periaortic tumors were administered pretherapeutic (131)I-tositumomab. Abdominal SPECT and whole-body planar images were obtained at 48, 72, and 144 h after tracer administration. Blood-pool activity concentrations were obtained from regions of interest drawn on the heart on the planar images. Tumor and blood activity concentrations, scaled to therapeutic administered activities-both standard and myeloablative-were input into a geometry and tracking model (GEANT, version 4) of the aorta. The simulated energy deposited in the arterial walls was collected and fitted, and the AD and biologic effective dose values to the aortic wall and tumors were obtained for standard therapeutic and hypothetical myeloablative administered activities. RESULTS: Arterial wall ADs from standard therapy were lower (0.6-3.7 Gy) than those typical from external-beam therapy, as were the tumor ADs (1.4-10.5 Gy). The ratios of tumor AD to arterial wall AD were greater for radioimmunotherapy by a factor of 1.9-4.0. For myeloablative therapy, artery wall ADs were in general less than those typical for external-beam therapy (9.4-11.4 Gy for 3 of 4 patients) but comparable for 1 patient (32.6 Gy). CONCLUSION: Blood vessel radiation dose can be estimated using the software package 3D-RD combined with GEANT modeling. The dosimetry analysis suggested that arterial wall toxicity is highly unlikely in standard dose radioimmunotherapy but should be considered a potential concern and limiting factor in myeloablative therapy.