FIRST MEDICAL FACULTY, CHARLES UNIVERSITY
INSTITUTE OF PHYSIOLOGY, CZECH ACADEMY OF SCIENCES
FACULTY OF MATHEMATICS AND PHYSICS, CHARLES UNIVERSITY
Branch: F4 - Physics of Molecular and Biological Structures
Abstract Booklet of Doctoral Thesis
BIOPHYSICAL MODELS OF NEURONS
Petr Marsálek
Advisor: Dr. Petr Lánský
Prague 1999
PRVNÍ LÉKARSKÁ FAKULTA UNIVERZITY KARLOVY
FYZIOLOGICKÝ ÚSTAV AKADEMIE VED CESKÉ REPUBLIKY
MATEMATICKO FYZIKÁLNÍ FAKULTA UNIVERZITY KARLOVY
Obor: F4 - Fyzika molekulárních a biologických struktur
Autoreferát doktorské disertacní práce
BIOFYZIKÁLNÍ MODELY NEURONU
Petr Marsálek
vedoucí práce: RNDr. Petr Lánský, CSc.
Praha 1999
= 4 cm
Uchazec /Candidate:
MUDr. Petr Marsálek
Ústav patologické fyziologie 1.LF UK
U nemocnice 5
128 53 Praha 2
= 4 cm
Cást výsledku pochází z pracoviste
/Part of the results comes from the:
Laboratory of Christof Koch, mail stop 139-74,
Computation and Neural Systems Program, Division of Biology,
CA 91125, Pasadena, California, USA
= 4 cm
Skolitel /Advisor:
RNDr. Petr Lánský, CSc.
Fyziologický ústav AVCR
Vídenská 1083
142 20 Praha 4
= 4 cm
Oponenti /Referees:
prof. RNDr. Vladislav Cápek, DrSc.
Fyzikální ústav UK
Ke Karlovu 5
121 16 Praha 2
doc. MUDr. Zdenek Wünsch, CSc.
Fyziologický ústav 1.LF UK
Albertov 2
128 00 Praha 2
Autoreferát byl rozeslán dne
/Submitted on April 23, 1999.
The defense date: June 30, 1999.
Contact addresses follow.
Obhajoba disertacní práce se koná ve stredu
30. cervna v 9 hodin
na MFF UK, Ke Karlovu 3,
121 16, Praha 2, v místnosti c. 105 (2. patro)
S disertacní prací je mozno se seznámit na Útvaru doktorandského studia MFF UK, Ke Karlovu 3, 121 16, Praha 2.
= 4 cm
Predseda oborové rady /Chairman of the subject board:
doc. RNDr. Otakar Jelínek, CSc.
Fyzikální ústav UK
Ke Karlovu 5
121 16 Praha 2
In the thesis, several problems from biophysics of neurons are addressed. Four types of neurons are studied: the pyramidal cells in neocortex, and in CA1 area in hippocampus, cells in the nucleus laminaris in avian auditory pathway, and the receptor cells in olfactory epithelium. Detailed and reduced models of these cells are used. Detailed models are based on the formalism of the Hodgkin-Huxley equations. Reduced models range from the stochastic two-point model, through the leaky integrator model to the simplest, perfect integrator model. The thesis is divided into parts 1-5. (1) In the first part we introduce all the mathematical tools for next parts and the basics of biology of neurons. (2) We study the dependence of the level of intracellular calcium on the previous firing in pyramidal cells. (3) Two-point stochastic model describes firing of olfactory cell in dependence on the odorant concentration. (4) Using several models we study spike timing in neocortex. In particular the question of the functional relationship between synaptic input jitter, and spike output jitter in individual pyramidal cells in neocortex. The term jitter means a standard deviation of events centered around some mean time. (5) Along with the discussion of results, several experimental methods, which are the sources of data for our models, are commented. The majority of results presented here were published in the following three papers: [1], [2] and [3].
Reprints of the abovementioned papers [1], [2] and [3] are included in the volume of the thesis. Abstracts of parts 1 and 5 were therefore written as paragraphs similar to already published abstracts of parts 2, 3 and 4. They are here for the purpose of better navigation in the thesis.
Basic equations are introduced. (1) Equations for the passive, capacitive and resistive properties for the neuronal membrane are shown. These are the equations for the RC-circuit, and the cable equation. We list the dimensions of all the quantities used in them. These two equations are linear, the first one is an ordinary, and the second one is a partial differential equation. (2) Equations for the active, membrane voltage dependent properties, are shown. These include the Hodgkin-Huxley equations (a system of nonlinear differential equations), two synaptic functions, alpha function and double exponential function, composed from a standard function of time, and an instance of a discontinuous function, used for the introduction of threshold behavior in reduced neuronal models. (3) The detailed, and the reduced models are introduced. Examples of them and of their particular implementation are given. (4) In a short review, biological information necessary for understanding the rest of the thesis is discussed. This includes the morphology of neurons, and the physiology of both the membrane potential and the calcium intracellular level. (5) The introductory part of the thesis is closed by comments to the current development in modeling neurons.
We modeled the influx of calcium ions into dendrites following active backpropagation of spike trains in a dendritic tree, using compartmental models of anatomically reconstructed pyramidal cells in a GENESIS program. Basic facts of ion channel densities in pyramidal cells were taken into account. The time scale of the backpropagating spike train development was longer than in previous models. We also studied the relationship between intracellular calcium dynamics and membrane voltage. Comparisons were made between two pyramidal cell prototypes and in simplified model. Our results show that: (1) Sodium and potassium channels are enough to explain regenerative backpropagating spike trains; (2) Intracellular calcium concentration changes are consistent in the range of milliseconds to seconds; (3) The simulations support several experimental observations in both hippocampal and neocortical cells. No additional parameter search optimization was necessary. Compartmental models can be used for investigating the biology of neurons, and then simplified for constructing neural networks.
Neurons need two basic properties to carry out their functions. The first is their ability to transduce the changes of the dendritic potential and to sum them in spatial and temporal dimensions. The second is their ability to elicit an action potential which can be transmitted along the axon at a long distance. This simulation study demonstrates how these two properties can be retracted to the two points of the neuron model. First we discussed the definition and general properties of the so-called two-point or spiking neuron model. Then a simple simulated solution of the first passage time problem of the birth and death process applied in this model was discussed. In case of olfactory cells, the model exhibited a behavior similar to the experimental data with parameter values corresponding to the suprathreshold concentrations of an odorant.
What is the relationship between the temporal jitter in the arrival times of individual synaptic inputs to a neuron and the resultant jitter in its output spike? We report that the rise time of firing rates of cells in striate and extrastriate visual cortex in the macaque monkey remain equally sharp at different stages of processing. Furthermore, as observed by others, multi-unit recordings from single units in the primate frontal lobe reveal a strong peak in their cross-correlation in the 10 - 150 msec range with very small temporal jitter (on the order of 1 msec). We explain these results using numerical models to study the relationship between the temporal jitter in excitatory and inhibitory synaptic input and the variability in the spike output timing in integrate-and-fire units and in a biophysically and anatomically detailed model of a cortical pyramidal cell. We conclude that under physiological circumstances, the standard deviation in the output jitter is linearly related to the standard deviation in the input jitter, with a constant less than one. Thus, the timing jitter in successive layers of such neurons will converge to a small value dictated by the jitter in axonal propagation times.
Since this work is a theoretical work, the discussion of experiments as sources of data was postponed into the closing part. Unlike in the classical physics, many experimental approaches to the measurement of a single biophysical value are yielding different results. The origin of such a difference is in the control of experimental conditions influencing the measurement. The better control of experimental conditions is allowed in recordings in vitro, mainly in brain slices, yet the conditions are less natural, compared to the in vivo recordings. In vivo recordings can be made either in the anesthetized, or even in the awake animal with chronically implanted recording and/or stimulating electrodes. Recording and stimulation can be extracellular and intracellular. Examples of all abovementioned techniques together with the procedure of obtaining the cell morphology are given in the closing part of the thesis. Results in computational models are discussed from the unifying point of view. Finally a list of selected open problems is given.
A typical neuron consists of dendrites (1), soma (or cell body) (2), axon (3), axonal branches (4), and axonal terminals (5). The signal processing in neuron proceeds in the same order, in which neuron parts were named here. Connections between neurons are made by synapses (6). All cellular structures consist of a lipid bilayer membrane with embedded, integral, proteins. from our viewpoint, the most prominent membrane proteins are ion channels. They are responsible for electrical properties of the membrane. Equations describing these properties can be found in [4]. Neurons process and transmit signals in form of membrane depolarization and hyperpolarization waves, or intracellular calcium ion changes. Depolarization of membrane is conveyed by dendrites (1) into the soma (2). When the depolarization in soma exceeds the voltage threshold, a characteristic uniform depolarization wave, called the spike, or the action potential, is emitted. The action potential is transmitted with no change, or processing, by axon (3) through axonal branches (4) into axonal terminals (5), ending by synapses (6). In synapses, an electrical event, presynaptic membrane depolarization, is transduced into a chemical signal, the chemical transmitter release. The transmitter then binds to the postsynaptic membrane giving rise to excitatory or inhibitory postsynaptic potential, (EPSP or IPSP). This way is a chemical signal transduced back to an electrical membrane event.
Further we will show several basic equations governing the behavior of the membrane. These equations are used in our models. The division of membrane properties into passive and active is borrowed from electrical engineering. Some elements of an electrical circuit, which cannot be described by linear differential equations like (1) and (3), are called active elements (for example transistors and diodes), compared to passive elements (resistors, capacitors, and inductances). The passive membrane is modeled by RC-circuit and the cable equation, the active membrane is modeled by the Hodgkin-Huxley equations, including synaptic currents.
A patch of the lipid bilayer membrane with basic protein pores only and without any voltage dependent ion channels can be described by the RC-circuit:
In an equivalent electrical model including the morphological description of a neuron, cell processes (dendrites, soma and axon) can be replaced by series of equivalent cylinders connected together. Individual cylinders are described by the cable equation:
Again, V is the voltage variable, and x is the space variable. I is input current, t is time. Rm, and Cmare the specific membrane resistance, and capacitance, and Riis the specific axial resistance of the cylinder. The se specific parameters have following dimensions: membrane capacitance [Cm]=Fm-2, membrane resistance m2 and axial resistance m. d is the diameter, and l is the length of the cylinder. Typical values of quantities in neurons are measured in mV, m, nA, ms and F. In particular, typical resistances are: the membrane resistance is in the range of kcm2, sometimes its inverse, conductance, in nScm-2 is used, axial resistance has a value of couple of hundreds cm-1
A patch of the membrane with voltage dependent ion channels is described by the Hodgkin-Huxley equations, [5]. As an example, the system describing firing in the avian nucleus magnocellularis cells is shown here:
where V, C, I and t are as in equation (1), GNa, GK and GL are maximal sodium (Na+), potassium (K+) ion channel, and ``leakage'' (L) conductances. VNa, VK, and VL are the reversal potentials. The second equation represents the system of four equations for activation and inactivation of voltage sensitive ion channels. Activation (m) and inactivation (h) variables are for both Na+ and K+ ions. They are denoted j=mNa, hNa, mK, hK. Their time constants are and steady state voltage dependences are given by the sigmoideal (Boltzmann) curves:
cm
| constant | units | m, Na | h, Na | m, K | h, K |
| mV | -40 | -45 | -54 | -50 | |
| Kj | mV | 3 | -3 | 6.5 | -6.5 |
| ms | 0.05 | 0.5 | 0.43 | 1.2 | |
| nS | 200 | 200 | 120 | 120 |
Depolarizing synapses are usually excitatory, and their effect is
called an excitatory postsynaptic potential, EPSP.
Hyperpolarizing synapses are usually inhibitory and their effect
is called an inhibitory postsynaptic potential, IPSP. A synaptic
response can be modeled by the following synaptic functions. The
alpha function is
.
Double
exponential function is
.
and
are time constants of
synaptic events. The alpha function has to be multiplied by a
constant with dimension [s-1] in order to get a
dimensionless function. Both the hyper- and de-polarizing action
of synapses is a conductance change for an appropriate ion:
We used both reduced and detailed models. Our two detailed models are L5, or neocortical pyramidal cell in Layer 5 and CA1, or hippocampal pyramidal cell in a structure Cornu Ammonis 1. The cable equation was solved by simulation packages GENESIS, [7], and NEURON, [8]. We used following reduced models: two-point stochastic models in [3], and perfect integrator, and leaky integrate-and-fire model in [2]. All these reduced models are introduced in [9].
The main source of data we use is experimental recording of neuron's electrical activity. Recording a train of individual action potentials, or spikes, can be made using intracellular, or extracellular electrodes. Intracellular recordings have lower noise, and about a decade or two higher voltage amplitude compared to extracellular recordings. For extracellular recordings it is enough to put the electrode tip close to the recorded cell. Both intra, and extracellular recordings can be used in both preparations, in brain slices, in vitro or in a whole animal, in vivo. One example: in [2] we used data, prepared as in [10].
For recording ionic currents, which we use in models, a special experimental setup is used, usually in brain slices. Data on ion channels kinetics are recorded using the patch clamp technique. The patch clamp technique is a miniature version of the classical voltage clamp experiment developed in [5]. Both methods allow V to be kept at a fixed value and to measure the opening and closing of ion channels in dependence on V. Data like those shown here in the Table are results of these measurements.
Optical scans of a cell in a fixed preparation are used for the computer aided reconstruction of a cell's 3D morphology. This morphology is then used in designing a detailed model from compartments of the measured length and diameter.
Results in the first paper [1] show how intracellular calcium is dependent on membrane potential. We proceeded further with a model, which explains how it is possible that action potential originates in the soma. We compared similarities among pyramidal cells in neocortex and hippocampus. We studied backpropagating spike train on intra-cellular calcium dynamics. Several experimental findings for apical dendrites were reproduced qualitatively in computer simulations. 1) The active backpropagation of action potential to the apical dendrite, 2) the dynamics of the intra-cellular calcium in the proximal part of the dendrite, 3) the accumulation of calcium along the apical trunk. We found that the backpropagation of trains of action potentials can be mediated by sodium and calcium channels along the apical dendrite. The accumulation of calcium depends mainly on the diameter of the dendrite. The concentration of sodium and potassium channels modulates the efficiency of the backpropagating train. The dynamics of instantaneous calcium concentration is different in the proximal and in the distal part. We have demonstrated in the model that all the phenomena observed in slices have a common denominator in the coupling of active, calcium and sodium channels in the dendritic tree. Our results unified in model different experimental results of stimulation and recording in brain slices.
In a paper [3] we investigated a stochastic model of neuron. This work was inspired by recordings of activity of olfactory cells in frogs. We did not use these data directly, however. The main result is a relationship, known elsewhere as F-I curve [4]. F-I curve is dependence of cell firing frequency (frequency of action potentials) on sustained (DC) input current. Quantities, further studied in this study, are: mean firing rate, its variance, together with these statistical values in the interspike interval (ISI). Other results describe analytical estimates of these quatities in dependence on a small set of discrete parameters of the model. The analytical estimates are supported by the simulations. This small set of discrete parameters is further compared to the classical Michaelis-Menten kinetics for the enzyme and substrate reaction. Olfactory cell is a receptor cell. In such a cell, the odorant concentration corresponds to the current injection in a general cell stimulation experiment. It is an ideal example for modeling chemical kinetics like events, and neuronal activity at the same time. Using equations like these for Michaelis-Menten reaction kinetics, we can reach the biophysically relevant level of description of a physiological event.
In a paper [2] we study the single cell response to the synaptic barrage. The term jitter means a standard deviation of events centered around some mean time. The basic propasal of the paper states the following. When EPSPs arrive to the neocortical cell around some mean time with a given jitter (input jitter), the time of the spike generated by the cell has a smaller time dispersion (output jitter). The function of dependence of these values, I/O jitter, and upper bound for it, are presented in the paper. This way, the precision of spike timing in neocortex can be preserved even in higher stages of cortical information processing. This result is shown in the three neuronal models with increasing complexity. In the simplest model, the perfect integrator, we have obtained a formula for I/O jitter by a calculation similar to that in the paper [3]. In more complex models we demonstrated our result using simulation.
Later on, following our paper, Feng [11] has shown some particular results on the way towards obtaining some analytical estimate for the perfect integrator model. He has found several solutions to special cases of our problem in the study of extremes of random processes. He substituted appropriate values to formulas found in this theory. He arrived to the same numbers as we did with our numerical simulations. This way he has shown by independent calculations that our numerical results in simplified models are correct. There is another idependent approach to the question of the neocortical processing speed. It can be summarized by stating that time constants known from nonlinear cable theory using Hodgkin-Huxley equations do not limit speed of processing in nervous system, [12].
In the Discussion part of the thesis, several questions are addressed. In particular, experimental methods for data acquisition, results of all three papers included in the thesis, and open questions for the future research are commented. For these topics, we refer to the thesis itself. Here we will list and comment a few literary sources. A general review and classification of detailed and reduced models can be found in [13]. A quantitative description of neural circuits at the subcellular level can be found in [14]. Physiological functions of neurons are described in [15]. A recent monograph on the single cell biophysics, including and citing our result from the paper [2] already, is [4]. A concise mathematical treatment of all the equations used throughout the thesis, is in [9]. In the paper [2], we used the most advanced simulation of detailed model of the whole thesis. We used a technique making detailed models more comparable to the in vivo situation, adding a synaptic background activity to the model, [16].
Numerical solving equations with a lot of parameters like those used in the thesis is far from elegant. When comparing a state of our knowledge of neuronal coding to the knowledge of coding in DNA, the lack of preciseness in neuroscience is striking. Collecting biophysical properties of different neurons can be compared to collecting X-ray diffraction snapshots of nucleic acid in the times before the discovery of its double helix structure. It is possible that the time of discovery of neuronal code is coming now, soon after the turn of the millennium. There are signs that, maybe, something interesting is going on. In individual experimental preparations, especially in lower animals, I/O function in single cell can be reconstructed using signal processing techniques. Linear, and even nonlinear kernels are reconstructed; after convolution of input with them, an output is obtained. Signal reconstruction algorithms can unify the theory of information processing in single neural cell by the same means, as the Hodgkin-Huxley equations did unify the theory of action potential fifty years ago. Or the Hodgkin-Huxley equations will show to be necessary prerequisite for generalizing signal reconstruction approach. Or maybe nothing from above mentioned possibilities will happen and the solution to the neuronal coding problem will come out using different techniques. In any case, many unanswered questions are still tempting our curiosity.
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