Intermittency and Diffusion in the Hodgkin-Huxley Model

Transcription

1 Intermittency and Diffusion in the Hodgkin-Huxley Model Gaspar Filipe Santos Magalhães Gomes Cano Thesis to obtain the Master of Science Degree in Engineering Physics Supervisor: Prof. Rui Manuel Agostinho Dilão Examination Committee Chairperson: Supervisor: Member of the Committee: Prof. Luís Filipe Moreira Mendes Prof. Rui Manuel Agostinho Dilão Prof. Maria Teresa Ferreira Marques Pinheiro April 2016

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3 Acknowledgments I would like to thank my supervisor, Prof. Rui Dilão. Without his ideas, insight, support and thorough supervision this work would not have been possible. I would also like to thank my family and all those who gave me their friendship and support while I worked on this thesis. i

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5 Abstract We show that action potentials in the Hodgkin-Huxley neuron model result from a type I intermittency phenomenon that occurs in the proximity of a saddle-node bifurcation of limit cycles. The restriction of the Hodgkin-Huxley equations to the center manifolds associated with the Hopf bifurcation points, subcritical and supercritical, can be described by the two-dimensional normal form of the codimension 2 Bautin bifurcation. For the Hodgkin-Huxley spatially extended model, describing the propagation of action potentials along axons, we show the existence of space propagating regular and chaotic diffusion waves, as well as type I spatial intermittency and a new type of chaotic intermittency. Chaotic intermittency occurs in the transition from a turbulent regime to the resting regime of the transmembrane potential and is characterised by the existence of a sequence of action potential spikes occurring at irregular time intervals. Keywords Hodgkin-Huxley model, type I intermittency, Bautin bifurcation, diffusion waves, chaotic intermittency. iii

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7 Resumo Mostra-se que os potenciais de acção no modelo neuronal de Hodgkin-Huxley resultam de um fenómeno de intermitência do tipo I, na vizinhança de uma bifurcação sela-nó de ciclos limite. A restrição das equações de Hodgkin-Huxley às variedades centrais associadas aos pontos de bifurcação de Hopf, subcrítico e supercrítico, pode ser descrita pela forma normal bi-dimensional da bifurcação de Bautin de codimensão 2. Para o modelo de Hodgkin-Huxley com componente espacial, que descreve a propagação de potenciais de acção ao longo de axónios, mostra-se que existem ondas de difusão regulares e caóticas, bem como intermitência espacial do tipo I e um novo tipo de intermitência caótica. A intermitência caótica ocorre na transição de um regime turbulento para um regime de repouso do potencial transmembranar e é caracterisada pela existência de uma sequêcia de potenciais de acção separados por intervalos de tempo irregulares. Palavras Chave Modelo de Hodgkin-Huxley, intermitência do tipo I, bifurcação de Bautin, ondas de difusão, intermitência caótica. v

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9 Contents 1 Introduction The Neuron The Hodgkin-Huxley Model The voltage-gated channels The Action Potential Thesis Outline The Diffusion free Hodgkin-Huxley model Analysis of dynamical stability The unstable knee The Bautin bifurcation Type I intermittency Diffusion in the Hodgkin-Huxley model Propagation of action potentials along the axon Dynamical stability and bifurcations Wave velocity and dispersion relation Type I spatial intermittency Chaotic Intermittency Conclusions 31 Bibliography 33 Appendix A Intermittency near a saddle-node limit cycle bifurcation A-1 Appendix B Stability of the steady state of the Hodgkin-Huxley equations B-1 vii

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11 List of Figures 1.1 Schematic of a regular neuron Cover of the 1963 Nobel Prize Programme, [1] Electrical circuit analogous to a cell membrane permeable to two different ionic species. Voltages E 1 and E 2, imposed by the batteries, result from the active transport done by ionic pumps Electrical circuit analogous to the Hodgkin-Huxley model, with channels for Na + and K +, a leakage channel R L for Cl and an input transmembrane current density i. Resistances R Na and R K are variable, since they represent the voltage-gated channels for sodium and potassium, respectively Electrical circuit showing the spatial component of the Hodgkin-Huxley model, where x is the direction of the axon length Left: membrane potential as a function of time. Right: two-dimensional (V, n) cut of the (V, n, m, h) phase-space. a) The perturbation felt by the neuron was not enough to generate an action potential, and it quickly falls back to its rest state. b) The perturbation was enough to produce an action potential response; the neuron fires Rate of open n, m and h gates, ionic current densities and membrane potential during an action potential Property 1) Voltage (a) and phase-space (b) response of the Hodgkin-Huxley (HH) equations (1.16) as a function of time, for the stimulation signal i(t) = 5 µa/cm 2 > I tr, during t = 35 ms, and i(t) = 0 otherwise. The impulse is above threshold. For i(t) = 5 µa/cm 2, t tr is 1.5 ms; for t 1 = 35 ms, I tr is 2.25 µa/cm Property 2) We perturbed the cell with a square wave, with a time difference between spikes t = 17.5 ms. For these parameter values, I tr = 2.25 µa/cm 2, t tr 1.5 ms and the refractory interval is t r 17.5 ms a) Bifurcation diagram for the diffusion free HH equations (2.1) at T = 6.3 C as a function of the transmembrane current density parameter i. saddle-node limit cycle (SNLC) bifurcation occurs for i = I 0 = 6.26 µa/cm 2, the subcritical Hopf bifurcation for i = I 1 = 9.77 µa/cm 2 and the supercritical Hopf bifurcation for i = I 2 = µa/cm 2. b) Variation of Hopf bifurcation points I 1 and I 2 with temperature. Calculated with XPPAUT ix

12 2.2 Two-dimensional cuts of the orbits of the limit cycles for a) i = 7 µa/cm 2 and b) i = 50 µa/cm 2 in equations (2.1). Full lines are stable cycles, dotted lines are unstable ones. The full dot in a) represents a stable fixed point, and the open dot in b) an unstable one a) Enlargement of the region in the neighbourhood of the knee that occurs in the interval [I 3, I 4 ] = [7.84, 7.92] µa/cm 2. b) Two-dimensional cut of the limit cycle orbits in phase-space for i = 7.88 µa/cm 2. Calculated with XPPAUT Disappearence of the one-to-many feature of the unstable limit cycle at T = 7.72 C. Calculated with XPPAUT Normal Bautin bifurcation parameters in cartesian (β 1,β 2 ) and polar (ρ,θ) coordinates, ( ) 4+ρ and the three different regions of equations (2.2). Border 1-2: θ = arccos SNLC bifurcation; border 2-3: subcritical Hopf bifurcation (H + ); border 3-1: supercritical Hopf bifurcation (H ) Bifurcation diagram for equations (2.2) as a function of the parameter θ = arctan( β 2 /β 1 ) Limit cycles and fixed points (red) in the three regions shown in figure 2.5, and trajectories (black) followed by the solutions of equations (2.2). Dashed circles are unstable limit cycles, while full circles are stable ones; full dots are stable fixed points, while hollow dots are unstable ones Membrane action potential response of the HH equations (2.1), for i = 6.20 µa/cm 2 < I 0 and initial conditions p 0 = (V (0) + V 0, n (0), m (0), h (0)). a) V 0 = 1.6 mv, b) V 0 = 1.7 mv, c) V 0 = 1.9 mv, and d) V 0 = 5.5 mv. The SNLC bifurcation occurs at i = I 0 = 6.26 µa/cm SNLC intermittency for the HH model. a) Logarithm of the number of spikes N of the action potential in the left vicinity of the SNLC bifurcation, as a function of the logarithm of ε = I 0 i. The slope of the fitted line is s = 0.505, in agreement with (2.3). b) Next amplitude map for the HH model, for the parameter value i = µa/cm 2, or ε = V N is the maximum value of the action potential spike number N. At the SNLC bifurcation, ε = 0, the parabolic profile shown touches the dotted line V N+1 = V N Spatial solutions of the HH equations (3.1), for D/ x 2 = 20 ms 1 at time t = 100 ms, for two different values of the transmembrane current density. In a), i 0 = 50 µa/cm 2 and the transmembrane potential converges to the steady state p (0) along the axon. In b), i 0 = 100 µa/cm 2 and the transmembrane potential develops a propagating periodic action potential Bifurcation diagram of the HH extended model (3.1), as a function of the transmembrane current density i 0 at the axon boundary and of the diffusion coefficient D/ x 2. We show two different types of intermittency, oscillations and spatial chaos (dark grey regions and dotted lines) ρ x

13 3.3 a) Period (and time intervals) of the oscillatory solutions of the HH extended model (3.1), as a function of the current density i 0, for the diffusion coefficient D/ x 2 = 20 ms 1. In the regions [I2, I3 ] (b) and [I4, I5 ] (c), the solutions of the HH extended model show chaos a) Wave vlocity of the action potential for D/ x 2 = 20 ms 1, for different values of i 0 within the parameter region [I1, I5 ]. b) Wave velocity as a function of parameter i 0 for five different diffusion coefficients of the HH equations Dispersion relation of action potential waves of the HH model, for five different values of the diffusion coefficient Type I intermittency for the HH extended model (3.1), with D/ x 2 = 20 cm 1. a) Logarithm of the number of spikes N of the action potential in the left vicinity of I1 = µa/cm 2. The slope of the fitted dotted line is s = 0.506, in good agreement with (2.3). b) Next amplitude map for the parameter value i 0 = µa/cm 2 or ε = V N (x = L/2) is the maximum value of the action potential spike number N, measured at the middle of the spatial domain [0, L]. The dotted line is the graph of the equation V N+1 = V N Chaotic intermittency for the HH extended model (3.1), with D/ x 2 = 20 cm 1. a) Logarithm of the number of spikes N of the action potential in the right vicinity of I5, for ε = i 0 I5 [0.001, 0.140]. The dotted line has slope s = 0.5. b) Next amplitude map for the parameter value i 0 = µa/cm 2 or ε = V N (x = L/2) is the maximum value of the action potential spike number N, measured at the middle of the spatial domain [0, L]. Chaotic intermittency does not show any scaling behaviour on the number of spikes as a function of ε. The dotted line is the graph of the equation V N+1 = V N xi

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15 Abbreviations HH Hodgkin-Huxley SNLC saddle-node limit cycle xiii

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17 1 Introduction Contents 1.1 The Neuron The Hodgkin-Huxley Model The Action Potential Thesis Outline

18 1.1 The Neuron In the nervous system, neurons are the excitable cells that are responsible for receiving stimuli and conducting impulses to other cells. They are able to transmit these signals over long distances across the body, and each can form up to tens of thousands of connections [2]. These connections are called synapses, and they can be both electrical and chemical [3]. Neurons work as the electrical wiring of the body, forming intricate and complex networks, responsible for our every thought and decision, movement and feeling. Just a regular human brain has dozens of billions of neurons [4, 5]. Figure 1.1: Schematic of a regular neuron. The diagram in figure 1.1 shows the three different parts that form a neuron. From left to right: the dendrites, which receive synaptic signals from other neurons or cells; the soma, the neuron body that contains the nucleus, responsible for the vitality of the cell; and the axon, which is the thinner cablelike structure, that extends from the soma, and is responsible for transmitting the signal generated by the neuron, and ends in the tree-shaped axon terminal, which transmits the propagated signal to other cells through synapses. Axons can have up to thousands of times in length the diameter of the soma. An electrical signal that propagates along the axon is called an action potential [2]. Each neuron behaves like a complex dynamical system, which can produce a variety of outcomes, according to the biochemical conditions it is subjected to, the signals it receives and the connections it makes. Neuronal networks can have computational properties, allowing for all kinds of calculations and outcomes, which is a field of study in itself (see, for example, [6]). In this thesis we shall focus merely on the dynamical properties of a single neuron-like cell. All cells have a potential difference between their inside and outside, known as the membrane potential. This potential difference arises from the difference in ionic concentrations on both sides of the membrane. Since cell membranes are lipid bilayers which are by themselves impermeable to ions, membranes possess multiple specialised channels and pumps made of proteins that are responsible for the transport of specific ions across the membrane. Pumps are responsible for moving ions against their concentration gradient, and for this they need to consume energy active transport. On the other hand, channels may open and close, but if open they simply let ions pass through them according to the concentration gradient, and therefore do not consume any energy passive transport. When 2

19 ions are exchanged across the cellular membrane there are two competing effects taking place: the difference in concentration in the two regions and the electrostatic forces between all the charged ions. This means that there is a complex system at work, regulating how the ions flow across the membrane while the channels are open. Channels can also be divided in two different categories gated and non-gated. Non-gated channels are always open, while gated channels open and close as a function of some variable (e.g. the membrane potential itself) [7]. Since each channel only lets one specific ionic species through, a cell membrane will be as permeable to some ion as the number of currently open channels for that ion. Each ion has a specific equilibrium state for inside and outside concentrations, which is given by its Nernst potential E ion = RT zf ln [ion] i [ion] o, (1.1) where E ion is the Nernst potential for some ion, R is the gas constant, T is the absolute temperature, F is the Faraday constant, z is the ion valence and [ion] represents the ionic concentration [4, 7]. In a neuron, the main ions responsible for the membrane potential are potassium (K + ), sodium (Na + ) and chloride (Cl ). Considering the electric field is constant across the membrane, and that all ions move independently, the Nernst equation can be generalized for the three most relevant ionic species: K +, Na + and Cl, giving the Goldman-Hodgkin-Katz (GHK) equation, [8]: V = RT F ln P K [K+ ] i + P Na [Na + ] i + P Cl [Cl ] o P K [K + ] o + P Na [Na + ] o + P Cl [Cl ] i, (1.2) where V is the potential difference between the outside and inside of the membrane and P is the permeability for each ionic species 1. Through this, one can calculate the membrane potential, just by knowing the ionic concentrations and their permeabilities. The stable membrane potential experienced when all the different ionic fluxes are in balance is called the resting potential. Since, at rest, the concentration of K + is much higher inside than outside the cell, while Na + and Cl have higher concentrations on the outside, the tendency will be for K + to leave the cell and for Na + and Cl to enter it. However, if no pumps existed, and ion channels were the only way ions could travel across the membrane, K + would tend to disappear from the cell and Na + and Cl would continuously flood its inside. Eventually there would be a loss in the gradients of ionic concentration that is necessary for the cell to maintain its resting potential. This is where the ion pumps come in, keeping the balance of ionic concentrations across the membrane in check. The main example of this is the Na-K pump that exchanges three Na + ions from the inside for two K + ions from the outside, being one of the most important ion transporters in cell membranes [7]. As we have just seen, the membrane potential in a neuron will depend on the ionic concentrations inside and outside the cell. However, these concentrations are dependent on the membrane potential itself, since the channels through which they pass open and close as a function of it. This means that the system is highly complex and one must take all of these factors into account, in order to describe it mathematically. It is in this description that the model developed by Hodgkin and Huxley was a pioneer, since it was the first to successfully describe and predict the behaviour of the membrane potential. In the following section we explain the Hodgkin-Huxley model in detail. 1 Membrane permeability is P = D/L, where L is the membrane width and D the diffusion coefficient for the species. 3

20 1.2 The Hodgkin-Huxley Model In 1952, Alan Lloyd Hodgkin and Andrew Huxley published a set of papers [9 12] where they explained in detail how they arrived at their successful description of the membrane potential behaviour in the giant axon of the squid Loligo. This revolutionary work earned them the Nobel Prize in Physiology or Medicine in 1963 (see figure 1.2). Figure 1.2: Cover of the 1963 Nobel Prize Programme, [1]. Hodgkin and Huxley were the first to measure the membrane potential in an axon. Since the squid giant axon can have up to one meter in length and one milimeter in diameter [13], they were able to apply a voltage-clamp technique which allowed them to successfully perform these measurements. In [9 12], they establish an analytical model that would describe how and why the potential behaved in the observed way. For this, they modelled the cell membrane as an electrical circuit. This circuit has three kinds of components: the ion channels, which act as resistors; the ion concentration gradients, acting as batteries; and the impermeable membrane, acting as a capacitor. outside i1 i 2 ic R 1 R 2 C m E 1 E 2 inside Figure 1.3: Electrical circuit analogous to a cell membrane permeable to two different ionic species. Voltages E 1 and E 2, imposed by the batteries, result from the active transport done by ionic pumps. 4

21 In figure 1.3, we show a schematic of an electrical circuit, with two different ionic channels, R 1 and R 2, for ionic species with Nernst potentials E 1 and E 2, and a capacitor C m, representing the membrane impermeability. Due to the effect of the ionic pumps that perform active transport across the membrane (like the Na-K pump), we can consider that the ionic gradients and the Nernst potential always remain the same. This is why we can represent them as a battery that imposes a constant voltage drop across the membrane. From figure 1.3 and Kirchhoff s circuit laws we get i C + i 1 + i 2 = 0, i 1 R 1 E 1 + (V o V i ) = 0, (1.3) i 2 R 2 E 2 + (V o V i ) = 0. where: the first equation comes from applying the junction rule to the top (outside) node; the second equation from applying the loop rule to the branches of R 1 and C m ; and the third from applying the loop rule to the branches of R 2 and C m. Considering the membrane potential to be V = V o V i, where the subscripts o and i distinguish the outside and inside of the cell membrane, respectively, we dv have i C = C m dt, and equations (1.3) become C m dv dt = g 1(V E 1 ) g 2 (V E 2 ), (1.4) where g = 1/R is the electrical conductance. Through this, it is possible to know how the membrane potential varies with time, just by knowing the membrane capacitance, the conductances of the two ionic channels, and the Nernst potentials of the ions passing through them. The actual circuit considered by Hodgkin and Huxley was a bit more complex than this one. They considered the membrane was permeable to three ionic currents: one for sodium, another for potassium and a leak current, mostly representing the flow of chloride ions. During their measurements of membrane potential response, they also introduced an external current into the membrane through a patch-clamp-like technique, to see how the potential would respond to certain stimuli. outside i mem C m i R Na R K R L E Na E K E L i mem inside Figure 1.4: Electrical circuit analogous to the Hodgkin-Huxley model, with channels for Na + and K +, a leakage channel R L for Cl and an input transmembrane current density i. Resistances R Na and R K are variable, since they represent the voltage-gated channels for sodium and potassium, respectively. 5

22 In figure 1.4, we show what the actual circuit looks like. The resistors for sodium and potassium are voltage-gated, so they are represented with an arrow across them. This means their resistance will vary as a function of the membrane potential itself. In section we explore this in detail. R L and E L represent the resistance and Nernst potential for the mentioned leak current, while i is the current density injected across the membrane (the one Hodgkin and Huxley were able to control during their experiments), and imem is the total current density crossing the membrane. Proceeding in the same way we did to get equation (1.4), we arrive at C m dv dt = i mem ḡ Na (V E Na ) ḡ K (V E K ) g L (V E L ) i, (1.5) where the variable conductances corresponding to the voltage-gated channels are represented by ḡ. If one wants to study electrical signal propagation in an axon, the spatial component also needs to be taken into account. For this, the neuron is considered as an electrical cable (as detailed, for example, in [14]) with one spacial dimension x, as seen in Figure 1.5. outside the membrane (V o ) i out R out R out i mem i mem inside the membrane (V i ) x i in R in x+dx R in Figure 1.5: Electrical circuit showing the spatial component of the Hodgkin-Huxley model, where x is the direction of the axon length. Following the current densities in figure 1.5 (analogous to what is done in [15]), if the ionic current densities inside and outside the cell membrane are considered to be ohmic, it arises by Ohm s law that V o (x + dx, t) V o (x, t) = i out R out, (1.6) V i (x + dx, t) V i (x, t) = i in R in, where R out and R in are the electrical resistances of the extra- and intracellular mediums. Taking the continuous limit, equations (1.6) become V o x = i out R out, V i x = i in R in. For any circuit node at position x, Kirchhoff s law gives which, again at the continuous limit, becomes i out (x, t) i out (x + dx, t) imem = 0, i in (x, t) i in (x + dx, t) + imem = 0, i out x = i mem, i in x = i mem. (1.7) (1.8) (1.9) 6

23 Taking the derivative of (1.7) together with equations (1.9), we get 2 V o x = i memr out, 2 V i x = i memr in, (1.10) and, since V = V o V i, (1.10) gives where D is the diffusion coefficient, 1 R out +R in. becomes D 2 V x 2 = i mem, (1.11) We can now replace this in equation (1.5), which C m dv dt = D 2 V x 2 ḡ Na (V E Na ) ḡ K (V E K ) g L (V E L ) i. (1.12) The voltage-gated channels We can now look at the ion channel conductances and how they depend on the membrane potential. Like we said above, the channels for sodium and potassium are voltage-gated, which means they have gates that will open or close, depending on the membrane potential they experience. Following [16], this model can be described by the diagram C α(v) O, (1.13) β(v) where C and O are the closed and open states of the gate, and α(v ) and β(v ) are voltage dependent rates at which the gate goes from one state to another. If we consider the fraction of open gates to be z, then 1 z is the fraction of closed gates, and from mass action law we get dz dt = α(v )(1 z) β(v )z. (1.14) There are two types of gates at work activation gates and inactivation gates. Their biochemical differences shall not concern us here, as we are merely interested in whether they are open or closed, and for this we only need to know their α(v ) and β(v ) functions. For the K + and Na + conductances, Hodgkin and Huxley proposed that ḡ K = g K n 4, ḡ Na = g Na m 3 h, (1.15) where n represents the probability of an activation gate for potassium to be open, while m and h represent the probability of an activation and inactivation gate for sodium to be open, respectively. The constants g K and g Na represent the maximum conductances for K + and Na +. For the ions to pass through a channel, all gates in that channel have to be open. Therefore, we can interpret (1.15) in the following way: if potassium channels have four activation gates, and n is the probability of one of them being open, then the total fraction of open potassium channels is n 4 ; if sodium channels have three activation gates, each with probability m of being open, and one inactivation gate with probability h of being open, then the total fraction of open sodium channels is m 3 h. Multiplying the fraction of open channels by the maximal conductance g, gives the conductance ḡ seen in (1.15). 7

24 From (1.12), (1.14) and (1.15) we arrive at the complete set of four coupled differential equations that make the Hodgkin-Huxley (HH) model: V C m t n t m t h t = D 2 V x 2 g Na m3 h(v E Na ) g K n 4 (V E K ) g L (V E L ) i, = α n (V )(1 n) β n (V )n, = α m (V )(1 m) β m (V )m, = α h (V )(1 h) β h (V )h. (1.16) Hodgkin and Huxley were then left with the problem of modelling the functions α and β for n, m and h. experimental data: They arrived at them heuristically, so that they would almost perfectly adjust to the V + 10 α n = 0.01φ e (V +10)/10 1, β n = 0.125φe V/80, V + 25 α m = 0.1φ e (V +25)/10 1, β m = 4φe V/18, α h = 0.07φe V/20 1, β h = φ e (V +30)/10 1, φ = 3 (T 6.3)/10, (1.17) where φ is the temperature coefficient that scales α and β for different temperature values. In these equations, the transmembrane potential V has units of mv, the transmembrane current density i that is applied to the cell is given in µa/cm 2 and time is measured in ms. In accordance with figures 1.4 and 1.5, positive values of i correspond to currents flowing from the outside into the cell. The model has been calibrated for the squid giant axon at the temperature T = 6.3 C, so that the resting potential when no current is applied (i = 0) is V = 0. The values of the constants are: C m = 1 µf/cm 2, g Na = 120 ms/cm 2, g K = 36 ms/cm 2 and g L = 0.3 ms/cm 2, where S=Ω 1 (siemens) is the unit of conductance. For this calibration, the Nernst equilibrium potentials are E Na = 115 mv, E K = 12 mv and E L = mv, [12]. For D = 0, the HH equations describe the potential drop across the walls of a globular cell and the ionic gradients inside the cell are negligible. Hodgkin and Huxley have estimated that the transmembrane diffusion coefficient is D = a/(2r 2 ), where a is the radius of the axon (considered as a cylinder) and R 2 is the specific resistivity along the interior of the axon. For the case of the squid giant axon, a = 238 µm, R 2 = 35.4 Ω cm and D = S, [12]. The electrophysiological state of any cell can be described by a HH type model, provided its electric state is controlled by the opening and closing of voltage sensitive channels. 1.3 The Action Potential The success of the HH model in describing the dynamics of certain types of neurons relies on its accuracy in reproducing experimental facts of patch-clamp experiments, including action potentials and their threshold properties, which we will now explain in detail. If we take D = 0 in equations (1.16), assuming the neuron is a globular cell without diffusion, its electrophysiological steady state is described by the vector quantity p (i) = (V (i), n (i), m (i), h (i)), 8

25 where the input transmembrane current density i is considered as an external parameter. When the neuron is at rest (i.e. i = 0, no transmembrane current is being received) its steady state p (0) is stable and is the unique limit set of the HH equations (1.16). These simple facts are well known and discussed in the literature, [17], [4] and [7], among others, and we explain them in chapter 2. Due to this stability in the vicinity of i = 0, small variations in the parameter i for a short period of time produce the same dynamic effects as changes in the initial conditions. In both cases, the electrophysiological state is displaced from rest, and we say the neuron has been perturbed. Its response to this perturbation can either be a small oscillation in phase-space that soon goes back to the rest state (figure 1.6a), or it can be a very characteristic large excursion through the phase-space (figure 1.6b). (Here, we are focusing exclusively on the simple (globular) cell with D = 0 in equations (1.16). In chapter 3 we look at how action potentials diffuse along the axon) a) V (mv) t (ms) n b) V (mv) t (ms) n Figure 1.6: Left: membrane potential as a function of time. Right: two-dimensional (V, n) cut of the (V, n, m, h) phase-space. a) The perturbation felt by the neuron was not enough to generate an action potential, and it quickly falls back to its rest state. b) The perturbation was enough to produce an action potential response; the neuron fires. This large excursion through the phase-space is called an action potential. When it is produced, the neuron is said to have fired. The distinction between both cases is quite clear, and there is a hard threshold between them. In figure 1.7 we look at what happens to the gates in the channels for sodium and potassium and how the ionic current densities behave during this action potential response. At t = 0, we have rest initial conditions p 0 = p (0). There are many more potassium channels open than sodium channels (m 3 0h 0 << n 4 0), which means the rest potential is mostly determined by the balance between the Nernst 9

26 -V ionic current (mv) (ma/cm 2 ) Na + K + rate of open gates h n m t (ms) Figure 1.7: Rate of open n, m and h gates, ionic current densities and membrane potential during an action potential. potential for potassium (E K = 12 mv) and chloride (E L = mv). Once the cell is perturbed (at t = 0, with a 5 µa/cm 2 current density i, lasting 5 ms), the membrane potential V is depolarized (becoming more negative), which causes the m gates to open very rapidly, bringing a large influx of sodium ions into the cell, and the membrane potential to become even more depolarized in a rapid fashion, almost reaching the Nernst potential for sodium (E Na = 112 mv). Simultaneously to this, the inactivation gates h for sodium have been closing during this depolarization, and the potassium activation gates n have been opening, expelling potassium ions from the cell. Both the rise of n and the decrease in h counter-balance the influence of the now almost completely open activation m gates. As V reaches its peak negative value, the m gates close very rapidly, making the potential rise very abruptly and an overshoot happens, bringing the potential close to the value of E K, higher than its rest state. During the time it takes for the potassium n gates and sodium h inactivation gates to recover their rest value, the cell is in its relaxation phase, in which it gradually recovers its rest value. During this recovery time, as we shall see ahead, the cell is much less prone to firing again. The ionic current densities in figure 1.7 are given by i ion = g ion (V E ion ). It is important to note that, according to the direction of the arrows in figure 1.3, these currents have been defined to be positive when they flow from the outside to the inside of the membrane. However, Na + and K + have a positive charge, and a positive inward current implies that positive ions are leaving the cell, while a negative inward current implies that positive ions are entering the cell. Thus, it follows that, during an action potential response, sodium ions enter the cell, while potassium ions leave it, which is consistent with 10

27 the description we made in section 1.1. The two main properties about action potentials and their associated threshold effect are the following: 1) Consider a cell or neuron at the steady state p (0). Imposing a current density to the cell during some short time t 1, there exists a time interval t tr and a threshold value I tr such that, if i(t) I tr, for t tr t m t tr, with finite real m > 1, and i(t) = 0 otherwise, the system develops a spike in its voltage response V (t) the action potential. Then, the voltage attenuates in time and the system returns to the stable steady state p (0). If i(t) < I tr, for t tr t m t tr, the voltage response V (t) attenuates in time to the stable steady state p (0) and the action potential does not develop - Figure 1.8 shows this action potential response for i(t) = 5 µa/cm 2 for t 1 = 35 ms. For this value of i, t tr is 1.5 ms; while for this t 1, I tr = 2.25 µa/cm a) 1.0 b) V (mv) V * (0) V * (5) 20 0 n x * (5) i (μa/cm 2 ) t (ms) 0.2 x * (0) V (mv) Figure 1.8: Property 1) Voltage (a) and phase-space (b) response of the HH equations (1.16) as a function of time, for the stimulation signal i(t) = 5 µa/cm 2 > I tr, during t = 35 ms, and i(t) = 0 otherwise. The impulse is above threshold. For i(t) = 5 µa/cm 2, t tr is 1.5 ms; for t 1 = 35 ms, I tr is 2.25 µa/cm 2. 2) Perturbing the HH equations (1.16) with a sequence of current density rectangular pulses above threshold I tr, and if the temporal differences between pulses are above some time interval t r, then the solutions of the HH equations develop the same number of pulses as in the current density exciting signal. We call t r the refractory interval - Figure 1.9. If we perturb continuously the HH model with a current density made of a sequence of rectangular pulses above threshold, and the temporal distance between pulses is below the refractory interval t r, then the number of pulses in the voltage response is smaller than the number of pulses of the exciting signal. In figures 1.8 and 1.9, we show the voltage response in a globular neuron ( D = 0) that was at rest (p 0 = p (0)) to a long current perturbation and to a square wave type perturbation. These two solutions illustrate the properties of the transient action potential behaviour as explained in the two cases described above the threshold property and the refractory property. 11

28 100 a) 1.0 b) V (mv) n i (μa/cm 2 ) t (ms) 0.2 x * (0) V (mv) Figure 1.9: Property 2) We perturbed the cell with a square wave, with a time difference between spikes t = 17.5 ms. For these parameter values, I tr = 2.25 µa/cm 2, t tr 1.5 ms and the refractory interval is t r 17.5 ms. 1.4 Thesis Outline One of the main goals of this thesis is to explain the origin of the threshold effect and the appearance of action potential spikes in the HH model equations and to characterise it dynamically. In chapter 2, we summarise the main dynamical properties of the solutions of the diffusion free ( D = 0) HH equations (1.16). It is shown that, for the parameter values described above and bifurcation parameter i, the asymptotic states of the HH four-dimensional equations (1.16) can be described by a codimension 2 Bautin bifurcation scenario. Upon variation of the bifurcation parameter i, this bifurcation scenario has a subcritical and a supercritical Hopf bifurcation and a global saddle-node bifurcation of limit cycles. We extend the concept of type I intermittency associated to saddle-node bifurcations of fixed points to intermittency of saddle-node bifurcation of limit cycles. We derive the scaling properties of this new type of intermittency. We show that action potentials, as observed in the HH equation, are originated by this new type I intermittency. In chapter 3, we analyse the spatially extended HH model ( D > 0) and show that the type I intermittency is responsible for the propagation of the action potentials along the axon. In the presence of diffusion, we find a new type I spatial intermittency, sustained oscillations develop, as well as chaotic or turbulent action potential propagation. For high values of transmembrane current densities, turbulent action potentials disappear and the convergence to a stable steady state occurs mixed with intermittent action potential chaotic spikes. We call chaotic intermittency to this new kind of intermittency of the HH extended model. The results presented in chapters 2 and 3 have been submitted for publication [18]. Finally, in chapter 4, we summarise the main conclusions of this work. 12

29 2 The Diffusion free Hodgkin-Huxley model Contents 2.1 Analysis of dynamical stability The Bautin bifurcation Type I intermittency

30 In this chapter we study Hodgkin-Huxley model without diffusion ( D = 0). We review its dynamical properties, and make a comparison with the normal form of the Bautin bifurcation. We show that the model has type I intermittency in the vicinity of a saddle-node limit cycle (SNLC) bifurcation, which results in an action potential response. 2.1 Analysis of dynamical stability As seen in equations (1.16), the diffusion free ( D = 0) HH equations take the form V C m t n t m t h t = g Na m 3 h(v E Na ) g K n 4 (V E K ) g L (V E L ) i, = α n (V )(1 n) β n (V )n, = α m (V )(1 m) β m (V )m, = α h (V )(1 h) β h (V )h. All of the variables and constants have been described in detail in section 1.2. For the analysis about to be made here, all the parameters are kept fixed with the exception of the current density i, which will be considered as the free parameter of the model, and the temperature T, which is always 6.3 C, except when indicated otherwise. (2.1) The basic bifurcation analysis as a function of i, including the existence of two Hopf bifurcations of fixed points, one subcritical and another supercritical, has been exhaustively analysed in [19] and [17]. For a review, recent references are [4] and [7]. The majority of authors have done the numerical analysis of the HH equations (1.16) with the bifurcation analysis software AUTO, [20], and XPPAUT, [21], which incorporates the former. bifurcation diagrams shown here have been calculated with XPPAUT. All -V (mv) 100 a) LC LC V * I I 0 I i (μa/cm 2 ) T (ºC) I 1 I i (μa/cm 2 ) b) Figure 2.1: a) Bifurcation diagram for the diffusion free HH equations (2.1) at T = 6.3 C as a function of the transmembrane current density parameter i. SNLC bifurcation occurs for i = I 0 = 6.26 µa/cm 2, the subcritical Hopf bifurcation for i = I 1 = 9.77 µa/cm 2 and the supercritical Hopf bifurcation for i = I 2 = µa/cm 2. b) Variation of Hopf bifurcation points I 1 and I 2 with temperature. Calculated with XPPAUT. In figure 2.1a, we show the bifurcation diagram for the diffusion free HH equations (2.1) at T = 6.3 C, as a function of the transmembrane current density parameter i. In it, we represent the stability of the coordinate V (i) of the fixed point p (i) and the maximum values of the V (i)-coordinate of the limit cycles (LC) associated with the Hopf bifurcations. Dotted lines correspond to unstable states 14

31 and continuous lines to stable ones. The SNLC bifurcation occurs for i = I 0 = 6.26 µa/cm 2, the subcritical Hopf bifurcation for i = I 1 = 9.77 µa/cm 2 and the supercritical Hopf bifurcation for i = I 2 = µa/cm 2. In figure 2.1b, we see how the position of the Hopf bifurcation points I 1 and I 2 varies with temperature. The two points coalesce at T = C, after which the fixed point p (i) remains stable for all values of i and no limit cycles appear. For temperatures below this point, the bifurcations and limit cycles are analogous to what is seen in 2.1a. The bifurcation diagram depicted in figure 2.1 summarises the main characteristics of the asymptotic solutions of the diffusion free ( D = 0) HH equations (2.1). These equations have a unique fixed point with coordinates p (i) = (V (i), n (i), m (i), h (i)), whose position in phase space depends on i (the other parameters in (2.1) are kept fixed). The fixed point p (i) has two Hopf bifurcations, one subcritical for i = I 1, and another supercritical for i = I 2. For i > I 2 and 0 i < I 1, the fixed point p (i) is stable. The local bifurcation analysis shows that for i < I 1 and I 1 i sufficiently small, the HH equations have at least two limit cycles, one stable and another unstable, [19] and [17]. The unstable limit cycle is created at I 1, for decreasing values of i, and the stable one is created at I 2, also for decreasing values of i. These two limit cycles collide at i = I 0 < I 1 and, for i < I 0 they do not exist. At i = I 0, the HH equation has a SNLC bifurcation, [22]. In figure 2.2, we show what the orbits of the limit cycles look like in the regions [I 0, I 1 ], where two limit cycles exist and [I 1, I 2 ], where only one exists. These are two-dimensional cuts of trajectories in the four-dimensional phase space (V, n, m, h) a) b) n n V (mv) V (mv) Figure 2.2: Two-dimensional cuts of the orbits of the limit cycles for a) i = 7 µa/cm 2 and b) i = 50 µa/cm 2 in equations (2.1). Full lines are stable cycles, dotted lines are unstable ones. The full dot in a) represents a stable fixed point, and the open dot in b) an unstable one. The fixed point p (i) is unstable for i in the interior of the interval [I 1, I 2 ] and stable outside. Away from the bifurcation points, the fixed point p (i) is hyperbolic. The overall behaviour of the bifurcation diagram in figure 2.1a can be understood as a codimension 2 Bautin or generalised Hopf bifurcation, [22] (in section 2.2 we cover this analogy in detail) The unstable knee At both I 3 = 7.84 µa/cm 2 and I 4 = 7.92 µa/cm 2 a saddle-saddle bifurcation of limit cycles appears, which produces a knee in the parameter region [I 3, I 4 ] in the unstable limit cycle. In figure 15

32 2.3a, we zoom in to this region to better show what it looks like in the bifurcation diagram, and in 2.3b we show the limit cycle orbits in a two-dimensional cut of the phase space inside [I 3, I 4 ]. We see LC a) b) -V (mv) LC n V * I 3 I I I i (μa/cm 2 ) V (mv) Figure 2.3: a) Enlargement of the region in the neighbourhood of the knee that occurs in the interval [I 3, I 4] = [7.84, 7.92] µa/cm 2. b) Two-dimensional cut of the limit cycle orbits in phase-space for i = 7.88 µa/cm 2. Calculated with XPPAUT. that we can have up to four limit cycles three unstable and a stable one. Strange phenomena have been reported in this very small region, such as the existence of chaotic behaviour ([23]), as well as a period doubling effect on the period of the limit cycles, (reported in [19] and [17]), however this effect is not a period doubling codimension 1 bifurcation. We can safely assume this does not affect the overall behaviour of the solutions of the HH equations, since it is restricted to the unstable limit cycle, for a very small region of i, and for T below 7.72 C. As we increase the temperature for values higher than this, the dotted limit cycle curve in figure 2.3a loses its one-to-many feature. 8 6 I 3 I 4 T (ºC) i (μa/cm 2 ) Figure 2.4: Disappearence of the one-to-many feature of the unstable limit cycle at T = 7.72 C. Calculated with XPPAUT. In figure 2.4, we can see the two points I 3 and I 4 coalescing and disappearing at T = 7.72 C. For temperatures higher than this, the unstable limit cycle behaves smoothly without further bifurcations. 16

33 2.2 The Bautin bifurcation The analysis done so far has shown that the Hodgkin-Huxley model has a generalized Hopf bifurcation, also called a Bautin bifurcation. We will now look at the Bautin bifurcation alone in more detail. Since equations 2.1 are so complex, it is easier to look at the normal form of the bifurcation, which is much simpler, and produces the same behaviour in only two dimensions (unlike the four needed in the HH model) and two parameters. The normal form of the Bautin bifurcation in cartesian coordinates can easily be obtained from its expression in polar coordinates (seen in Appendix A, equations (A.1)), and is { ẋ = β 1 x y + β 2 x ( x 2 + y 2) x ( x 2 + y 2) 2, ẏ = x + β 1 y + β 2 y ( x 2 + y 2) y ( x 2 + y 2) 2, (2.2) where β 1 and β 2 are real parameters. Figure 2.5 shows an analysis of these parameters and for which values the different stages of the Bautin bifurcation occur. β 2 2 SNLC 3 H + 1 θ 0 -β 1 ρ H - Figure 2.5: Normal Bautin bifurcation parameters in cartesian ( (β 1,β 2) and polar (ρ,θ) coordinates, and the ) three different regions of equations (2.2). Border 1-2: θ = arccos - SNLC bifurcation; border 2-3: 4+ρ 2 2 ρ subcritical Hopf bifurcation (H +); border 3-1: supercritical Hopf bifurcation (H ). The horizontal axis of figure 2.5 represents the symmetric of the parameter β 1, and the vertical axis represents β 2. In order to better characterise the Bautin bifurcation, it is more intuitive to describe these parameters in terms of their polar equivalent coordinates ρ and θ, that, as seen in the figure, have been chosen to be ρ = β1 2 + β2 2 and θ = arctan( β 2/β 1 ). For β 1 = β2/4, 2 which is the ( ) 4+ρ2 2 same as saying, for θ = arccos there is a saddle node bifurcation in the radial variable ρ r = x 2 + y 2, which corresponds to a SNLC bifurcation in the cartesian coordinates (x, y). At θ = π/2 there is a subcritical Hopf bifurcation, and at θ = 3π/2 there is a supercritical Hopf bifurcation. Since the behaviour of the system is similar for all values of ρ, we can fix it at ρ = 1 and study the dynamical behaviour of the system as a function of the parameter θ This is shown in figure

34 1 x 0.5 SNLC LC LC 0 x * 0 π 2 π θ 3 π 2 2 π Figure 2.6: Bifurcation diagram for equations (2.2) as a function of the parameter θ = arctan( β 2/β 1). Comparing the three different regions in figure 2.5 with the bifurcation diagram of figure 2.6, we can see that in region 1, θ [0, cos 1 ( 5 2)] [3π/2, 2π], the fixed point p = (x, y ) is stable and there are no limit cycles; in region 2, θ [cos 1 ( 5 2), π/2], the fixed point remains stable, and there are two limit cycles, one stable and another unstable; and in region 3, θ [π/2, 3π/2], the fixed point is unstable and there is only one limit cycle, which is stable. The unstable limit cycle originates at the subcritical Hopf bifurcation at θ = π/2, the stable limit cycle at θ = 3π/2, and both collide ( ) at the SNLC bifurcation, at θ = cos 1 4+ρ 2 2 ρ. This is the exact same behaviour we observed in figure 2.1a for the Hodgkin-Huxley model, now for a much simpler system of equations. In figure 2.7 we show what the limit cycles (in red) in the three regions of figure 2.5 look like, and the black arrows represent the trajectories followed by solutions of the normal form of the Bautin bifurcation. It has just been shown that the response of the Hodgkin-Huxley model to changes in the parameter i is exactly the same as the response of the normal form of the Bautin bifurcation for the parameter θ = arctan ( β 2 /β 1 ). 2.3 Type I intermittency We have just established in detail that, just like in region 1 in the normal form of the Bautin bifurcation, for i < I 0, the HH equations only have one stable fixed point, and no limit cycle exists. However, for this region, these equations develop an action potential, resembling the limit cycle response only found in other parameter regions. In fact, one can now realise that figures 1.6b, 1.8 and 1.9 were examples of this. It is this phenomenon that we now explain in this section. If a cell or neuron is perturbed with some constant transmembrane current density i < I 0, where I 0 is the parameter value of the SNLC bifurcation, and there is no diffusion ( D = 0), the asymptotic time solutions of the HH equations converge to the stable fixed point p (i), for any initial condition away from it. In this parameter region(i < I 0 ), if the initial condition is away from the fixed point, let us say p 0 = (V 0, n 0, m 0, h 0 ) p (i), then the response of the system has two possible outcomes. 18

35 y 0.0 y x x y x Figure 2.7: Limit cycles and fixed points (red) in the three regions shown in figure 2.5, and trajectories (black) followed by the solutions of equations (2.2). Dashed circles are unstable limit cycles, while full circles are stable ones; full dots are stable fixed points, while hollow dots are unstable ones. If p 0 is close enough to p (i), then the asymptotic solutions of the HH equations converge to p (i), without ever doing a long excursion through phase space regions away from the fixed point. On the contrary, if p 0 is sufficiently displaced form p (i) in the (V, n, m, h) four-dimensional phase space, the solution of the HH equations does a large excursion in phase space, resembling, during some time, an almost periodic orbit action potential response. These two regions in phase space are separated by a boundary or threshold. As the parameter i approaches I 0 and p 0 is sufficiently displaced from p (i), the larger are the number of transient spikes that appear in the potential V. For the same i < I 0, it is possible to produce zero, one, or more spikes, depending on how far p 0 is from p (i). There is, however, an upper limit for which, no matter how much we continue to displace p 0 from p (i), no more spikes are produced. Thus, there is a maximum number of obtainable spikes for each i < I 0. Depending on the value of p 0, the number of spikes generated must be either equal to or below this maximum. In figure 2.8, we show this transient behaviour of the solutions of equations (2.1), for several values of p 0 and fixed i < I 0. As shown in figure 2.8, the solutions of the HH equations are action potential type responses and, as we shall see now, they are the result of a type I intermittency phenomenon, [24], [25] and [26], 19