From Single Cell to Neural Field - Dynamics on Different Scales

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(3) F ROM S INGLE C ELL TO N EURAL F IELD D YNAMICS ON D IFFERENT S CALES. K OEN D IJKSTRA.

(4) The research presented in this thesis was carried out at the group of Applied Analysis, the faculty of Electrical Engineering, Mathematics and Computer Science, and MIRA Institute for Biomedical Technology and Technical Medicine, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands.. This work was supported by the Twente Graduate School (TGS Award 2013).. Dijkstra, Koen. From Single Cell to Neural Field - Dynamics on Different Scales. Ph.D. Thesis, University of Twente, 2017. Copyright © 2017 by Koen Dijkstra. Cover design by Joscha Thelosen. Printed by Gildeprint. ISBN : 978-90-365-4319-4. DOI : 10.3990/1.9789036543194.

(5) FROM SINGLE CELL TO NEURAL FIELD DYNAMICS ON DIFFERENT SCALES. PROEFSCHRIFT. ter verkrijging van de graad van doctor aan de Universiteit Twente, op gezag van de rector magnificus, prof. dr. T.T.M. Palstra, volgens besluit van het College voor Promoties in het openbaar te verdedigen op vrijdag 24 maart 2017 om 14:45 uur. door. Koen Dijkstra geboren op 6 juni 1987 te Delmenhorst.

(6) Dit proefschrift is goedgekeurd door de promotoren Prof. dr. S.A. van Gils Prof. dr. ir. M.J.A.M. van Putten en de copromotor Prof. dr. Yu.A. Kuznetsov.

(7) Samenstelling van de promotiecommissie: Voorzitter en secretaris: Prof. dr. P.M.G. Apers. Universiteit Twente. Promotoren: Prof. dr. S.A. van Gils Prof. dr. ir. M.J.A.M. van Putten. Universiteit Twente Universiteit Twente. Copromotor: Prof. dr. Yu.A. Kuznetsov. Universiteit Twente. Leden: Prof. dr. R.J. Boucherie Prof. dr. A. Doelman Prof. dr. W. van Drongelen Prof. dr. ir. B.J. Geurts Dr. J. Hofmeijer Prof. dr. P.H.E. Tiesinga. Universiteit Twente Universiteit Leiden University of Chicago Universiteit Twente Universiteit Twente Radboud Universiteit Nijmegen.

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(9) Acknowledgements Even though there is only one name on the cover of this thesis, a lot of people contributed to its contents. I want to seize this opportunity to thank everyone who directly or indirectly helped me in arriving at this point. First and foremost, I am grateful to my doctoral advisors. Stephan, without your encouragement, I would have never started this project. As my daily supervisor, you gave me the freedom that I needed and were always there for support. Michel, your enthusiasm is contagious and inspiring. After every meeting I was full of new ideas and motivated to keep going. Yuri, thanks for always asking the right questions and teaching me basically everything I know about bifurcation theory. Next, I want to thank all other co-authors who I had the pleasure of working with. Jeannette, your input greatly improved the quality of Chapter 2, and I learned a lot from your clear and precise writing style. Wim, Tahra, and Christoph, I highly enjoyed working on our joint project that forms the basis of Chapter 4. Sid, your previous work and supervision during my Master’s laid the foundation of Chapter 5, and I would have never been able to write down the finer mathematical details of sun-star calculus without the help of Sebastiaan. I am also very grateful to the other members of my graduation committee, Richard Boucherie, Arjen Doelman, Bernard Geurts, and Paul Tiesinga for the valuable time they spent to read and evaluate this thesis. From May to July 2015, I visited Steven Schiff at Penn State University. I thank him for his hospitality, and Andrew and Karma for making my time in the States unforgettable. A big thank you goes to all current and former colleagues in Twente, in particular from the Department of Applied Mathematics, who created a very pleasant working atmosphere. I really enjoyed the many entertaining discussions during group meetvii.

(10) viii. Acknowledgements. ings and at the lunch table, playing for our own futsal team Pi Hard, the pub quizzes, and the many other non-work related activities highlighted by two vacations at the house of Felix in the Austrian mountains. So thanks to Anton, Bettina, Bijoy, Christoph, Daniela, Edo, Felix, Gijs, Gjerrit, Hil, Huan, Ivana, Jurgen, Kamiel, Laura, Leonie, Matthias, Mihaela, Milos, Nastya, Nishant, Paolo, Sjoerd, Wilbert, Yoeri, and all others not named here. The secretaries Linda and Mariëlle deserve a special mention for their assistance in booking flights, solving administrative issues, and arranging everything else. Furthermore, I want to thank my old friends from my hometown Kleve, who often provided a welcome distraction from work. I am particularly indebted to Joscha for designing the marvelous cover of this thesis, and to Marie for proofreading parts of the final manuscript. Of course, I am solely responsible for all remaining errors. I am honored that my my brother Jelle and my friend and colleague Wilbert have agreed to be my paranymphs. Thanks that you are willing to support me on the day of my promotion. Finally, a special thank you goes to my wonderful parents for their love and support throughout the years. Henk en Myriam, thanks for always giving me the freedom to follow my own path in life.. Koen, February 2017.

(11) Contents. 1. A Brief Introduction to Mathematical Neuroscience . . . . . . . . . . . . . . . . . . . . . 1 1.1 Neuronal Biophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 The Membrane Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 The Hodgkin-Huxley Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.3 Synaptic Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Neural Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 A Heuristic Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 Propagation Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. 2. A Biophysical Model for Cytotoxic Cell Swelling . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Neuron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Fast Transient Na+ and Delayed Rectifier K+ Current . . . . . . . . . . 2.2.3 Voltage-Gated Cl- Current Through SLC26A11 . . . . . . . . . . . . . . . 2.2.4 Specific Leak Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Na+ /K+ -ATPase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 KCl Cotransport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.7 Intracellular Concentrations and the Membrane Potential . . . . . . 2.2.8 Cell Volume and Water Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.9 Model Parameter Estimation and Validation . . . . . . . . . . . . . . . . . . 2.2.10 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Intracellular Osmolarity is Essentially Defined by [Cl- ]i . . . . . . . . 2.3.2 Osmotic Pressure in Gibbs-Donnan Equilibrium . . . . . . . . . . . . . . 2.3.3 Ion Permeabilities Determine Speed of Neuronal Swelling . . . . . 2.3.4 Cell Volume Critically Depends on Remaining Pump Activity .. 15 16 17 17 18 20 20 21 21 22 22 23 23 24 24 24 28 28. ix.

(12) x. Contents. 2.4. 2.3.5 Functionality is not Restored at Physiological Pump Strengths . 30 2.3.6 Na+ Channel Blockers May Reverse Cytotoxic Edema . . . . . . . . . 31 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33. 3. A Rate-Reduced Neuron Model for Complex Spiking Behavior . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Single Spiking Neuron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Fast Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Spiking Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Neuronal Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Rate-Reduced Neuron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Frequency Response of the Rate-Reduced Model . . . . . . . . . . . . . . 3.3.2 The Spiking Rate Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Example: Augmenting a Neural Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37 38 38 40 41 42 46 46 49 51 52. 4. The Cross-Scale Effects of Neural Interactions during Seizure Activity . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Patients and Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Macroscopic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Mesoscopic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Interaction between Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.6 Synthetic LFP Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Analysis of the MEA Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Spike Triggered Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Decomposition of Spike Triggered Averages . . . . . . . . . . . . . . . . . . 4.3.4 Multi-Scale Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 The Role of Feedforward Inhibition . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 55 56 57 57 59 59 60 60 61 62 62 63 65 68 68 72. 5. Local Bifurcations in Neural Field Models with Transmission Delays . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Functional Analytic Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Spectral Properties of the Linearized Problem . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Eigenvalues and Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 The Resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Example: Two Pitchfork-Hopf Bifurcations . . . . . . . . . . . . . . . . . . .. 75 76 77 82 82 85 88.

(13) Contents. xi. 5.4. Normal Forms on the Center Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.4.1 The Critical Center Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.4.2 The Canonical Pitchfork-Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . 94 5.4.3 Normal Form Coefficients and their Practical Computation . . . . 95 5.4.4 Example Continued: Actual Normal Form Coefficients . . . . . . . . 99 5.5 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.5.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.5.2 Example Continued: Bifurcation Analysis and Simulations . . . . 100 5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6. Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Samenvatting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.

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(15) C HAPTER 1. A Brief Introduction to Mathematical Neuroscience “When we talk mathematics, we may be discussing a secondary language built on the primary language truly used by the central nervous system.” – John von Neumann. 1.1 Neuronal Biophysics The elementary processing unit in the central nervous system is called a neuron. A human brain contains billions of neurons, and each of them is connected to thousands of other neurons, yielding trillions of connections. The basic morphology of a neuron consists of three main parts: a cell body, called the soma, dendrites, and an axon (Figure 1.1). The dendrites form a tree-like structure that acts as an input device: incoming signals are integrated and transferred to the soma, where they are processed in a nonlinear fashion. The tube-like axon of a neuron plays the role of an output device: it transmits electrical signals to other neurons. Compared to the dendrites, which normally end close to the cell body, the axon can stretch over several centimeters, forming long-range connections to other areas of the brain.. 1.1.1 The Membrane Potential Like other living cells, neurons are surrounded by a semipermeable membrane. It contains channels through which specific ions can freely diffuse, and transporter and pump proteins that actively move ions through the membrane and establish ion concentration gradients (Figure 1.2). Ion channels are electrically equivalent to a set of batteries and resistors inserted in the membrane, and therefore create a potential 1.

(16) 2. 1 A Brief Introduction to Mathematical Neuroscience Dendrite. Axon Cell body. Figure 1.1: Caricature of a neuron. Dendrites receive incoming signals from the terminal ends of other axons. The axon divides into many branches, allowing for a signal to pass simultaneously to many target cells.. difference between the intra- and extracellular side. Under physiological conditions, the resting membrane potential of neurons is about −60 mV to −80 mV with respect to the exterior of the cell, with significant gradients between the intra- and extracellular concentrations [X]i and [X]e of the main permeant ions X ∈ {Na+ , K+ , Cl- , Ca2+ } (Table 1.1). The hypothetical membrane potential at which the electrical driving force on an ionic species X is equal and opposite to the diffusive driving force acting on X is called the Nerst potential of X (Nernst, 1888), and is given by EX =. RT [X]e ln , zX F [X]i. (1.1). where zX is the valence of ion X, and F, R, and T are Faraday’s constant, the universal gas constant and the absolute temperature, respectively. In general, the ionic current IX through membrane channels depends on the shape of the electric field within the membrane. When the electric field is assumed to be constant, IX is given by the Goldman-Hodgkin-Katz current equation (Goldman, 1943; Hodgkin and Katz, 1949) Table 1.1: Typical concentrations of membrane-permeant ions. Na+. K+. Cl-. Ca2+. Intracellular concentration. 110 mM. 145 mM. 137 mM. 10−5 mM. Extracellular concentration. 150 mM. 143 mM. 135 mM. 1 mM. Ionic species.

(17) 1.1 Neuronal Biophysics. 3 Na+ /K+ pump. Ion channels Na+. K+ 2K+. Extracellular. Lipid bilayer. Intracellular. Na+. K+. 3Na+ ATP ADP+Pi. Figure 1.2: Constituents of the neuronal membrane. The cell membrane is a lipid bilayer that forms an impermeable barrier for ions. Ion channels are proteins that form a pore through the membrane, allowing only specific ions to pass through. Ion pumps, in this example the Na+ /K+ pump, use energy from the hydrolysis of ATP into ADP and a phosphate ion to move ions against their electrochemical gradients. Adapted from Sterratt et al. (2011, Figure 2.2). z FV z2X F2 [X]i − [X]e exp − XRT , IX = PX V FV RT 1 − exp − zXRT. (1.2). where V is the membrane potential and PX denotes the permeability of the membrane to ion X. Clearly, the ion channel current IX is zero if and only if the membrane potential equals the Nernst potential EX . As the Nernst potentials are typically different from the membrane potential, channel currents of permeant ions do not vanish. This implies that ion gradients as observed in physiological conditions, and the associated resting membrane potential, can only be maintained by active ion transport by energydependent pumps. In situations where energy supply is insufficient to fuel the pumps, ion gradients will be reduced and corresponding Nernst potentials will change. Eventually, an equilibrium is reached where all individual ion channel currents are zero: the Gibbs-Donnan equilibrium. The membrane voltage at Gibbs-Donnan equilibrium, the Gibbs-Donnan potential, is typically non-zero and results from the presence of large, negatively charged proteins that are unable to pass through the semipermeable membrane.. 1.1.2 The Hodgkin-Huxley Model Neurons are excitable cells capable of generating action potentials when electrically stimulated. Action potentials, also known as ‘spikes’, are short-lasting events in which the (local) electrical potential between the interior and exterior of the cell quickly rises.

(18) 4. 1 A Brief Introduction to Mathematical Neuroscience. B 50. Fraction of open channels. Membrane potential (mV). A. 10 60% -30. -70 50. 150. 250. Time (ms). m3 h n4 40% -70 20%. 0%. 350. C. 50. 52. 54. 56. 10. 50. Time (ms). D 9. 1. h∞. m∞. Time constant (ms). Equilibrium value. 60%. 2 3. 40% 1 3. n∞. τh. 6 40%. τn. 3. τm 0. 0 -70. -30. 10. 50. Membrane potential (mV). -70. -30. Membrane potential (mV). Figure 1.3: Action potentials and gating dynamics in the Hodgkin-Huxley model. (A) Neuronal action potentials. (B) Fraction of the open sodium (blue) and potassium (orange) channels during the time course of a single action potential (indicated by a dotted line). (C) Equilibrium values of the sodium activation (blue), sodium inactivation (green), and potassium activation (orange) gating variable as a function of the membrane potential. (D) Corresponding time constants as a function of the membrane potential.. and falls, following a consistent trajectory (Figure 1.3A). In a seminal series of articles, Hodgkin and Huxley (1952a,b,c,d,e) were the first to model neural excitability and action potential generation mathematically. Their work won them a share of the 1963 Nobel Prize in Physiology or Medicine, and their model is still one of the most widely used models in neuroscience. The semipermeable cell membrane that separates the interior of the cell from the extracellular liquid and can be modeled as a capacitor in parallel with an ionic current, resulting in.

(19) 1.1 Neuronal Biophysics. 5. dV + Iion = Iapp , dt. (1.3). C. where C is the capacitance of the membrane, Iion is the total ionic current through the membrane, and Iapp is an externally applied current. By using an experimental method called the voltage clamp, Hodgkin and Huxley were able to measure the ionic currents through the membrane of the giant axon of a squid, while holding the membrane potential at a set level. They determined that the total ionic current in the giant axon of a squid could be subdivided into three components: a sodium current, a potassium current, and a leak current that is primarily carried by chloride ions. In order to explain their experimental data, they hypothesized that the membrane permeabilities1 of sodium and potassium changed dynamically as a function of the membrane potential. In the Hodgkin Huxley model, these voltage-gated permeabilities are modeled by the assumption that sodium and potassium channels contain a number of gates that can be either in an open or closed state, and that a channel is permeable if and only if all of its gates are open. The probability q of a certain gate to be in the open state is then assumed to evolve according to the first order kinetic equation dq = αq (V )(1 − q) − β q (V )q, dt. (1.4). where αq and β q are voltage-dependent opening and closing rates, respectively. Gates that open upon membrane depolarization are called activation gates, and gates that close upon membrane depolarization are called inactivation gates. Sodium channels are modeled with three activation gates, denoted by m, and one inactivation gate, denoted by h. Potassium channels are modeled with four activation gates, denoted by n, and no inactivation gates. The total membrane permeabilities of sodium and potassium can hence be approximated by PNa+ = m3 hPNa+ , PK+ = n4 PK+ ,. (1.5). where PNa+ and PK+ denote maximal permeabilities, and where m3 h and n4 can be interpreted as the fraction of open sodium and potassium channels, respectively (Figure 1.3B). For a fixed (‘clamped’) membrane potential V, the gating variables q ∈ {m, h, n} will eventually reach their equilibrium value (Figure 1.3C) q ∞ (V ) = 1. α q (V ) . α q (V ) + β q (V ). (1.6). In their original model, Hodgkin and Huxley assumed that ion channel currents were ohmic, and they hence modeled conductances rather than permeabilities..

(20) 6. 1 A Brief Introduction to Mathematical Neuroscience. The time course for approaching this steady state is given by a simple exponential with time constant (Figure 1.3D) τq (V ) =. 1 . α q (V ) + β q (V ). (1.7). Combining (1.3), (1.4), and (1.5) yields the Hodgkin-Huxley equations  dV     dt     dm    dt  dh     dt     dn   dt. =−. 1 Iion (V, m, h, n) − Iapp , C. = αm (V )(1 − m) − β m (V )m,. (HH). = αh (V )(1 − h) − β h (V )h, = αn (V )(1 − n) − β n (V )n,. where Iion = INa+ + IK+ + ICl− . It is important to note that ion concentrations are assumed to be constant in the Hodgkin-Huxley equations, even though ion pumps are not modeled explicitly. The Hodgkin-Huxley model is therefore only applicable in situations where ion homeostasis is maintained.. 1.1.3 Synaptic Currents The junction where the axon of a sending neuron connects to a dendrite of a receiving neuron is called a synapse. The sending neuron is usually referred to as the presynaptic neuron, and the receiving neuron is referred to as the postsynaptic neuron. At a synapse, the arrival of a presynaptic spike at time t = T triggers the release of neurotransmitter into the synaptic cleft, which binds to receptors on the postsynaptic dendrite (Figure 1.4). This in turn causes ion channels on the postsynaptic side to open and therefore leads to a conductance change of the postsynaptic membrane. The synaptic current Is induced by this single spike can hence be written as Is (t) = gs (t − T ) Es − V (t) ,. (1.8). where V (t) denotes the postsynaptic membrane potential at time t, Es is the reversal potential of the synapse, and the function gs : R 7→ R models the time course of the change in conductance. The synaptic current (1.8) can be simplified by assuming that the membrane potential of a neuron stays close to its resting potential most of the time. The arrival of a spike can then be modeled to directly induce a current rather than a conductance change, such that.

(21) 1.1 Neuronal Biophysics. 7. Presynaptic axon blubNeurotransmitter Synaptic cleft. Receptor. Postsynaptic dendrite. Figure 1.4: Structure of a typical (chemical) synapse. Following the arrival of an action potential, neurotransmitter is released into the synaptic cleft, where it binds to receptors on the postsynaptic dendrite.. Is (t) = cs hs (t − T ),. (1.9). where cs is the total charge flowing into the postsynaptic neuron as a result of a single spike of the presynaptic neuron. The magnitude of cs measures the strength of the synapse, and the sign of cs determines whether the synapse is excitatory (cs > 0) or inhibitory (cs < 0). The function hs : R → R represents the normalized characteristic time course of the synaptic current, satisfying hs ( t ) = 0. for. t ≤ 0,. hs ( t ) ≥ 0. for. t > 0,. and. Z∞. hs (t) dt = 1.. (1.10). 0. Typically, the characteristic time course hs is given by a superposition of exponentials. A common choice is the double exponential hs ( t ) =. αs β s e − αs t − e − β s t H ( t ), β s − αs. (1.11). 1 where H is the Heaviside step function and αs−1 and β− s are synaptic time constants that determine the rise and decay time (Figure 1.5). Taking the limit βαss → 0 in (1.11) yields the simple exponential decay. h s ( t ) = α s e − αs t H ( t ), and taking the limit. αs βs. (1.12). → 1 in (1.11) yields the alpha function. hs (t) = α2s te−αs t H (t).. (1.13).

(22) 8. 1 A Brief Introduction to Mathematical Neuroscience. A. B. C. 1 20. 0. 1 10. hs (t ) (ms−1 ). 1 10. hs (t ) (ms−1 ). hs (t ) (ms−1 ). 1 10. 1 20. 0 0. 20. 40. t (ms). 1 20. 0 0. t (ms). 40. 0. t (ms). 40. Figure 1.5: Characteristic time courses of the synaptic current. (A) Simple exponential decay (1.12) with αs−1 = 10 ms. (B) Double exponential (1.11) with αs−1 = 9 ms 1 −1 and β− s = 1 ms. (C) Alpha function (1.13) with αs = 5 ms.. Now let us look at a population P = {1, . . . , N } of labeled neurons. Assuming that multiple synaptic responses sum linearly, the total synaptic input ui to neuron i ∈ P is given by ui ( t ) =. N. ∑ cij ∑. j =1. m ∈Z. hij (t − Tjm ),. (1.14). where Ti1 , Ti2 , . . . are the spiking times of neuron i ∈ P, and where cij and hij denote the strength and characteristic time course of the j → i synaps, respectively.. 1.2 Neural Fields When large populations of neurons are modeled by networks of individual, interconnected cells, the high dimensionality of state and parameter spaces makes mathematical analysis impossible and numerical simulations costly. Moreover, large network simulations provide little insight into global dynamical properties. A popular modeling approach to circumvent the aforementioned problems is the use of neural field equations. These models aim to describe the dynamics of large neuronal populations, where spikes of individual neurons are replaced by (averaged) spiking rates and space is continuous. The first neural field model can be attributed to Beurle (1956), however, the theory really took off with the work of Wilson and Cowan (1972, 1973), Amari (1975, 1977), and Nunez (1974). Another advantage of neural fields is that they are often well suited to model experimental data. In brain slice preparations, spiking rates can be measured with an extracellular electrode, while intracellular recordings are much more involved..

(23) 1.2 Neural Fields. 9. Furthermore, the most common clinical measurement techniques of the brain, electroencephalography (EEG) and functional magnetic resonance imaging (fMRI), both represent the average activity of large groups of neurons and may therefore be better modeled by population equations.. 1.2.1 A Heuristic Derivation Let us take another look at our population P = {1, . . . , N } of interconnected neurons, and define the spike train si of neuron i ∈ P by si ( t ) =. ∑. m ∈Z. δ t − Tim ,. (1.15). where δ denotes the Dirac delta function. This enables us to write the total synaptic current (1.14) as a sum of convolutions ui ( t ) =. N. ∑ cij (hij ∗ s j )(t).. (1.16). j =1. Furthermore, we define the spiking rate ri of neuron i ∈ P as a weighted temporal average of its spike train si , i.e. ri (t) = (w ∗ si )(t),. (1.17). where w : R 7→ R is a (normalized) weight kernel with support in an interval of length ∆t containing 0. If ∆t is small compared to the time scale of ui , the total synaptic current can be assumed to be approximately constant on intervals of length ∆t, and filtering (1.16) with w yields ui ( t ) =. N. ∑ cij (hij ∗ r j )(t).. (1.18). j =1. The central assumption in the derivation of neural field equations is that the spiking rate of a neuron is given by its instantaneous input, such that r i ( t ) = Si u i ( t ). . (SRA). for all i ∈ P and some spiking rate function Si : R 7→ R. Hence we arrive at the closed set of equations.

(24) 10. 1 A Brief Introduction to Mathematical Neuroscience. ui ( t ) =. N. ∑ cij. j =1. hij ∗ (S j ◦ u j ) (t). for all. i ∈ P.. (1.19). Now assume that the characteristic time course hij only depends on the postsynaptic neuron i, such that we can write hij (t) = hi (t). for all. i, j ∈ P.. (1.20). This is a reasonable assumption if synapses are fast, such that the decay rate of the synaptic current is dominated by the time constant of the postsynaptic membrane. Furthermore, we assume that hi is the impulse response of a linear differential operator Li , i.e. (. L i h i ( t ) = δ ( t ), h i (0) = 0. for all. i ∈ P.. (1.21). For the characteristic time course given in (1.11), the corresponding differential operator Li is given by. Li =. . 1 d 1+ αi dt. . 1 d 1+ . β i dt. (1.22). Assumption (1.20) and (1.21) enable us to transform the integral equations (1.19) into a set of differential equations given by. Li ui ( t ) =. N. ∑ cij Sj. j =1. u j (t). . for all. i ∈ P.. (NMa). On the other hand, if synapses are slow compared to the membrane time constant, it is reasonable to assume that the characteristic time course hij only depends on the presynaptic neuron j, i.e. hij (t) = h j (t). for all. i, j ∈ P.. (1.23). If we then define the activity ai of neuron i ∈ P as a i ( t ) = h i ∗ ( Si ◦ u i ) ( t ) ,. (1.24). we can use (1.23) and (1.24) to transform the integral equations (1.19) into a set of differential equations given by.

(25) 1.2 Neural Fields. L i a i ( t ) = Si. 11. . N. ∑ cij a j (t). j =1. . for all. i ∈ P.. (NMb). It is always possible to partition our total population P into M subpopulations P1 , . . . , P M , such that neurons within each subpopulation share the same characteristic synaptic response and spiking rate function. If we let pi ∈ {1, . . . , M } denote the subpopulation of neuron i, we can assume without loss of generality that and. Li = L pi. Si = S p i. for all. i ∈ P.. (1.25). Furthermore, lets assume that for all k ∈ {1, . . . , M }, all neurons belonging to subpopulation k are equidistantly placed on some bounded and connected spatial domain Ωk ⊂ Rd for some d ∈ {1, 2, 3}, and let xi ∈ Ω pi denote the position of neuron i ∈ P. In this case, every neuron is uniquely determined by its position and subpopulation, and we can therefore write cij =. 1 Jp p (x , x ) ρ pj i j i j. (1.26). for some connectivity function Jkl : Ωk × Ωl 7→ R that describes how subpopulation l influences subpopulation k. The constant ρk denotes the neuron density of subpopulation k, which is given by ρk =. |Pk | . |Ωk |. (1.27). By using (1.25) and (1.26), we can rewrite (NMa) and (NMb) as. Lk ui ( t ) =. M. |Ω |. ∑ |Pll | ∑. j ∈Pl. l =1. Jkl ( xi , x j )Sl u j (t). . (1.28a). and. L k a i ( t ) = Sk. . M. |Ω | ∑ |Pll | l =1. ∑. j ∈Pl. Jkl ( xi , x j ) a j (t). . (1.28b). for all k ∈ {1, . . . , M} and i ∈ Pk , respectively. The two systems (1.28a) and (1.28b) describe the evolution of a discrete network of neurons. By looking at it from another perspective, these sets of equations could also be interpreted as spatial discretizations of the continuous neural field equations. Lk uk (t, x ) =. M Z. ∑. l =1. Ωl. Jkl ( x, x 0 )Sl ul (t, x 0 ) dx 0. (NFa).

(26) 12. 1 A Brief Introduction to Mathematical Neuroscience. and. Lk ak (t, x ) = Sk. . M Z. ∑. l =1. Ωl. 0. 0. Jkl ( x, x ) al (t, x ) dx. 0. . (NFb). for all k ∈ {1, . . . , M }, respectively, where uk (t, x ) and ak (t, x ) are the postsynaptic current and presynaptic activity of subpopulation k at time t and position x ∈ Ωk . The neural field (NFa) is often referred to as the voltage-based model (although in our case, ‘current-based’ would be more appropriate), and neural field (NFb) is called the activity-based model. Finally, neural fields in which the spatial extension is neglected are called neural masses. If one assumes that the spatial dependence in (NFa) and (NFb) is negligible, one arrives at the neural mass equations (NMa) and (NMb), where subscripts now correspond to subpopulations rather than individual neurons.. 1.2.2 Propagation Delays In the derivation of the neural field equations (NFa) and (NFb) we have neglected any delays that arise due to dendritic integration and finite propagation speeds of action potentials along axons. Incorporating these delays into (NFa) and (NFb) yields the delayed neural field equations. Lk uk (t, x ) =. M Z. ∑. l =1. Ωl. Jkl ( x, x 0 )Sl ul (t − τkl ( x, x 0 ), x 0 ) dx 0. (DNFa). and. Lk ak (t, x ) = Sk. . M Z. ∑. l =1. Ωl. Jkl ( x, x 0 ) al (t − τkl ( x, x 0 ), x 0 ) dx 0. . (DNFb). for all k ∈ {1, . . . , M }, respectively, where τkl ( x, x 0 ) measures the time it takes for a signal sent by a type-l neuron located at position x 0 ∈ Ωl to reach a type-k neuron located at position x ∈ Ωk . A natural choice for the delay function τkl : Ωk × Ωl 7→ R≥0 is given by τkl ( x, x 0 ) = χkl +. |x − x0 | , νkl. (1.29). where χkl is a fixed delay caused by synaptic processes and dendritic integration, and νkl is the propagation speed of action potentials along axons..

(27) 1.3 Thesis Outline. 13. 1.3 Thesis Outline In Chapter 2, we extend the classical Hodgkin-Huxley formalism (HH) with dynamic ion concentrations and osmotically driven volume changes. This enables us to study the ionic mechanisms of cytotoxic edema in a biophysical single neuron model. Cytotoxic edema most commonly results from energy shortage, such as in cerebral ischemia, and refers to the swelling of brain cells as a result of water entering from the extracellular space. We show that the principle of electroneutrality explains why chloride influx is essential for the development of cytotoxic edema. Using numerical bifurcation analysis, we show that a tipping point of the energy supply exists, below which the cell volume rapidly increases. We simulate realistic time courses to and reveal critical components of neuronal swelling in conditions of low energy supply. Furthermore, we show that, after transient blockade of the energy supply, cytotoxic edema may be reversed by temporary blockade of sodium channels. In Chapter 3, we focus on the spiking rate assumption (SRA). While it is very convenient and widely used, it only applies to simple neurons whose spiking rate can be well approximated by their instantaneous input. In reality, however, many neurons have dynamic spiking thresholds, and, in general, spiking rates therefore also depend on past input. Common examples are spike frequency adaptation and rebound spiking, with the former describing the phenomenon that spiking rates in some neurons decrease over time when they are stimulated by a constant input, and the latter referring to the property of some neurons to spike after being released from inhibitory input. We present a simple phenomenological neuron model that can mimic a wide variety of these biologically realistic spiking patterns. The model is constructed in a way that makes a spiking rate-reduced version readily available. It can then be easily applied to existing neural field models of both type (NFa) and (NFb). With the help of an example, we finally show that the resulting augmented neural field equations closely capture the complex spatiotemporal dynamics of a large network of spiking neurons. In Chapter 4, we analyze and model the interaction between neuronal dynamics across network scales. Specifically, we analyze how seizure activity at a mesoscopic wavefront (measured with a mm-sized Utah microelectrode array) interacts with the macroscopic, surrounding networks (recorded with a cm-sized ECoG electrode array) in human patients. We then model these seizure dynamics by coupling a neural field of type (NFb), representing the scale observed by the Utah array, to a neural mass of type (NMb), representing the activity observed by ECoG electrodes. We show how macroscopic properties can affect the frequency and amplitude of the seizure’s oscillations. Additionally, we show how one neuronal function, i.e. feedforward.

(28) 14. 1 A Brief Introduction to Mathematical Neuroscience. inhibition, plays different roles across scales: inhibition at the mesoscopic wavefront fails and allows seizure activity to propagate, but at the macroscopic scale, inhibition of the surrounding territory is activated via long-range intracortical connections and creates a distinct pathway to a postictal state. Ultimately, our modeling framework can be employed to examine meso-macroscopic perturbations, and thereby to evaluate strategies to promote transition to a postictal state. Finally, in Chapter 5, we study bifurcations in neural field equations with transmission delays. In particular, we study spectral properties of the delayed neural field equation (DNFa) in the most simple case M = 1 and Ω = (−1, 1). We demonstrate how symmetry arguments can be used to simplify the computation of eigenvalues for a specific, yet very general form of the connectivity, and show how normal form coefficients can be evaluated with the help of residue calculus. The proposed methods are illustrated by an extensive study of two particular pitchfork-Hopf bifurcations..

(29) C HAPTER 2. A Biophysical Model for Cytotoxic Cell Swelling Abstract We present a dynamical biophysical model to explain neuronal swelling underlying cytotoxic edema in conditions of low energy supply, as observed in cerebral ischemia. Our model contains Hodgkin-Huxley type ion currents, a recently discovered voltage-gated chloride flux through the ion exchanger SLC26A11, active KCC2-mediated chloride extrusion and ATP-dependent pumps. The model predicts changes in ion gradients and cell swelling during ischemia of various severity or channel blockage with realistic timescales. We theoretically substantiate experimental observations of chloride influx generating cytotoxic edema, while sodium entry alone does not. We show a tipping point of Na+ /K+ -ATPase functioning, where below cell volume rapidly increases as a function of remaining pump activity, and a GibbsDonnan-like equilibrium state is reached. This precludes return to physiological conditions even when pump strength returns to baseline. However, when voltagegated sodium channels are temporarily blocked, cell volume and membrane potential normalize, yielding a potential therapeutic strategy† .. † This Chapter is adapted from K. Dijkstra, J. Hofmeijer, S.A. van Gils, and M.J.A.M. van Putten. A biophysical model for cytotoxic cell swelling. Journal of Neuroscience, 36:11881-11890, 2016.. 15.

(30) 16. 2 A Biophysical Model for Cytotoxic Cell Swelling. 2.1 Introduction Cerebral edema is classically subdivided into cytotoxic and vasogenic edema (Kempski, 2001; Klatzo, 1987; Simard et al., 2007). Vasogenic edema originates from a compromised blood-brain barrier and accumulation of water in the extracellular space due to entry of osmotically active particles from the vasculature. This is observed in a variety of conditions, such as traumatic brain injury, tumors or ischemia (Donkin and Vink, 2010). Otherwise, cytotoxic edema most commonly results from energy shortage, such as in cerebral ischemia, and arises from swelling of neurons or astrocytes due to a redistribution of extracellular fluid to the intracellular compartment. While this does not lead to tissue swelling, it generates the driving force for the movement of constituents from the intravascular space into the brain, which does cause tissue swelling (Simard et al., 2007). Formation of cerebral cytotoxic edema is often recognized as an important pathophysiological mechanism leading to initial neuronal damage or secondary deterioration in patients with cerebral ischemia (Stokum et al., 2016). In this population, interindividual differences with regard to the speed and degree of cytotoxic edema formation are large and not well understood (Hofmeijer et al., 2009). A better understanding of key processes involved in cytotoxic edema formation, and potential explanations for the large variability in extent and time course observed in the clinic, may help to identify patients at risk and assist in defining potential targets for intervention. Since the vast majority of osmotically active particles in the brain are ions (Somjen, 2004), the study of cytotoxic edema is essentially the study of maladaptive ion transport (Stokum et al., 2016). We here introduce a dynamical biophysical model to identify fundamental determinants of cytotoxic cell swelling and to simulate its development. Following the approach originally introduced for cardiac cells (DiFrancesco and Noble, 1985) and ion concentration dynamics during seizures and spreading depression in neurons (Kager et al., 2000), we extend the Hodgkin-Huxley formalism (Hodgkin and Huxley, 1952e) to include dynamic intracellular ion concentrations and resulting volume dynamics. Our model allows quantitative predictions of cytotoxic cell swelling with regard to its timescale, severity, and relation to the availability of energy. In the original Hodgkin-Huxley equations, ion concentrations and corresponding Nernst potentials are assumed to be constant. This assumption holds in physiological conditions where ATP supply is sufficient to maintain ion homeostasis and firing rates are modest. However, it loses its validity if ATP supply does not meet its need (Zandt et al., 2011, 2013a), as observed in ischemia (Stokum et al., 2016), or pathological brain states which are intrinsically characterized by a massive redistribution of ions, such as seizures (Fröhlich et al., 2008; Raimondo et al., 2015).

(31) 2.2 Materials and Methods. 17. and spreading depolarization (Somjen, 2001; Zandt et al., 2013b). Recently, a similar approach was used in a three compartment model to study cell swelling in astrocytes and neurons during spreading depolarization (Hübel and Ullah, 2016). We focus on neuronal swelling only and assume that extracellular ion concentrations are constant, yielding a single compartment model. While this is clearly not in agreement with biological reality, it closely resembles conditions in brain slice experiments. Furthermore, all ion fluxes in the model are purely biophysical, and corresponding parameters can therefore be measured directly in experiments. The model explains why the intracellular chloride levels and remaining activity of ATP-dependent pumps are major determinants of cytotoxic edema. Our simulations demonstrate that at a critical value of pump activity the cell volume strongly increases, and a pathological Gibbs-Donnan-like equilibrium state is reached. Neurons in this state do not recover if the Na/K pump activity returns to baseline or even beyond. However, subsequent temporary blockade of sodium channels provokes reversal of cytotoxic edema and functional recovery. This may explain why blockers of voltagegated sodium channels have shown to prevent neuronal death in various experimental models of cerebral ischemia (Lynch et al., 1995; Carter, 1998), and may assist in better patient selection for therapeutic strategies.. 2.2 Materials and Methods 2.2.1 Neuron Model Our model neuron consisted of a single intracellular compartment with a variable volume separated from the extracellular solution by a semi-permeable cell membrane (Figure 2.1). The extracellular bath was assumed to be infinite, such that all properties of the extracellular space were constant model parameters (Table 2.1). The model T ), a delayed rectifier potassium current contained a transient sodium current ( INa + L ), potassium ( I L ) and chloride ( IKD+ ), and specific leak currents for sodium ( INa + K+ L ( ICl- ). It was recently shown that the ion exchanger SLC26A11 is highly expressed in cortical and hippocampal neurons (Rahmati et al., 2013), acts as a chloride channel that is opened by depolarization of the membrane, and plays an important role in cell G ) modeled this swelling (Rungta et al., 2015). An additional voltage-gated current ( ICl chloride flux (Figure 2.2). Physiological intracellular resting concentrations were maintained by Na+ /K+ -ATPase, which generated a net transmembrane current ( IPump ), and an electroneutral KCl cotransporter, which generated a molar transmembrane ion flux ( JKCl ). The cell volume changed due to osmotically-induced water flux ( JH2 O )..

(32) 18. 2 A Biophysical Model for Cytotoxic Cell Swelling. IPump. JKCl. Extracellular Bath [Na+ ]e = 152.0 mM [ K+ ] e =. 3.0 mM. -. [Cl ]e = 135.0 mM [A- ]e. = 20.0 mM. T INa +. IKD+ G ICl -. [Na+ ]i = 10.0 mM [ K+ ] i -. = 145.0 mM 7.0 mM. IKL+. = 148.0 mM. L ICl -. [Cl ]i = [A- ]i. L INa +. JH2 O Figure 2.1: Schematic model overview with typical ion concentrations. Negatively charged, impermeant macromolecules are denoted by A- . Leak and voltage-gated ion channels (yellow) yield ion currents which are balanced by the electrogenic ATP-dependent Na+ /K+ pump (cyan) and the electroneutral KCl cotransporter (orange). While the pump moves both Na+ and K+ against their electrochemical gradients and therefore needs ATP to run, the KCl cotransporter uses the energy stored in the standing transmembrane gradient of K+ to move Cl- out of the cell. Any difference in osmolarity between the intra- and extracellular space will yield a water flux across the membrane (blue), changing the cell volume.. 2.2.2 Fast Transient Na+ and Delayed Rectifier K+ Current The kinetics of the transient sodium and delayed rectifier potassium current were based on a model by Kager et al. (2000), T INa +. IKD+. [Na+ ]i − [Na+ ]e exp − FV RT = , 1 − exp − FV RT + + FV 2 D 2 F V [K ]i − [K ]e exp − RT , = PK+ n RT 1 − exp − FV RT 2 T 3 F V PNa +m h RT. (2.1) (2.2). T D where V was the membrane potential, PNa + and PK+ were maximal membrane permeabilities, and F, R and T were Faraday’s constant, the universal gas constant and the . absolute temperature, respectively (Table 2.1). The variables q ∈ m, h, n were the usual Hodgkin-Huxley gates: sodium activation, sodium inactivation, and potassium activation, respectively. They evolved according to. dq = αq (1 − q) − β q q, dt with voltage-dependent opening rates αq and closing rates β q (Table 2.2).. (2.3).

(33) 2.2 Materials and Methods. 19. Table 2.1: Model constants, parameters and variables with default (resting) values. Constant. Value. Description. F. 96485.333 C/mol. Faraday’s constant. R. 8.3144598 (C V)/(mol K). Universal gas constant. T. 310 K. Absolute temperature. Parameter. Value. Description. C. 20 pF. Membrane capacitance. T PNa + L PNa + PKD+ PKL+ G PCl L PCl -. 800 µm3 /s. Maximal transient Na+ permeability. 2 µm3 /s. Leak Na+ permeability. 400 µm3 /s. Maximal delayed rectifier K+ permeability. 20 µm3 /s. Leak K+ permeability. 19.5 µm3 /s. Maximal voltage-gated Cl- permeability. 2.5 µm3 /s. Leak Cl- permeability. QPump. 54.5 pA. Maximal Na+ /K+ pump current. UKCl. 1.3 fmol/(s V). KCl cotransporter strength. 152 mM. Extracellular bath Na+ concentration. 3 mM. Extracellular bath K+ concentration. [Cl ]e. 135 mM. Extracellular bath Cl- concentration. LH2 O. 2 µm3 /(s bar). Effective membrane water permeability. [S]e. 310 mM. Total extracellular solute concentration. NiA. 296 fmol. Intracellular amount of impermeant anions. Variable. Value. Description. +. [Na ]e +. [K ]e -. −65.5 mV. V. Membrane potential. m. 0.013. Transient Na+ activation gate. h. 0.987. Transient Na+ inactivation gate. n. 0.003. Delayed rectifier K+ activation gate. [Na+ ]i. 10 mM. Intracellular Na+ concentration. [ K+ ] i. 145 mM. Intracellular K+ concentration. [Cl ]i. 7 mM. Intracellular Cl- concentration. W. 2000 µm3. Intracellular volume. -.

(34) 20. 2 A Biophysical Model for Cytotoxic Cell Swelling Table 2.2: Opening and closing rates of gating variables (Kager et al., 2000).. Term. Expression. Discription. 0.32(V + 52 mV) kHz/mV mV 1 − exp − V +4 52 mV. αm. 0.28(V + 25 mV) kHz/mV mV exp V +5 25 −1 mV V + 53 mV kHz 0.128 exp − 18 mV. βm αh. 4 kHz mV 1 + exp − V +5 30 mV. βh. 0.016(V + 35 mV) kHz/mV mV 1 − exp − V +5 35 mV V + 50 mV 0.25 exp − kHz 40 mV. αn βn. Opening rate transient Na+ activation gate Closing rate transient Na+ activation gate Opening rate transient Na+ inactivation gate Closing rate transient Na+ inactivation gate Opening rate delayed rectifier K+ activation gate Closing rate delayed rectifier K+ activation gate. 2.2.3 Voltage-Gated Cl- Current Through SLC26A11 Gating of the chloride current through the ion exchanger SLC26A11 was assumed to be instantaneous and given by a sigmoidal function of the membrane potential, fitted to experimental data of Rungta et al. (2015) (Figure 2.2), G ICl. F2 V [Cl ]i − [Cl ]e exp = 10 mV RT 1 − exp FV 1 + exp − V + RT 10 mV G PCl -. FV RT. . ,. (2.4). G was the maximal gated chloride permeability. where PCl -. 2.2.4 Specific Leak Currents The sodium, potassium and chloride leak currents were modeled as regular Goldman-Hodgkin-Katz currents (Hille, 2001) with fixed leak permeabilities PXL for . X ∈ Na+ , K+ , Cl- , IXL. =. FV − [X]e exp − zXRT . FV 1 − exp − zXRT. z2 F 2 [X]i PXL X V RT. (2.5). Both the sodium and chloride permeability were low compared to the potassium permeability (Table 2.1). The total leak current was fitted to experimental data from coronal brain slices of rats (Rungta et al., 2015, Figure 7C)..

(35) DIDS sensitive current (pA). 2.2 Materials and Methods. 21. 375. Control Low [Cl- ]e. 250. 125. 0 -40. -20. 0. Membrane potential (mV). 20. Figure 2.2: Voltage-gated chloride current through the ion exchanger SLC26A11. Marks denote voltage clamp recordings of the transmembrane current blocked by application of DIDS in coronal brain slices of rats, published in Rungta et al. (2015, Figure 7D). Error bars represent SEM. Raw data were kindly provided by the Brian MacVicar lab. Dashed lines depict the modeled G for normal [Cl- ] = 135 mM and low [Cl- ] = 10.5 mM voltage-gated chloride current ICl e e extracellular and corresponding resting intracellular chloride concentrations.. 2.2.5 Na+ /K+ -ATPase In each cycle, the Na+ /K+ - ATPase exchanges three intracellular sodium ions for two extracellular potassium ions, and therefore generates a net transmembrane current IPump . The net pump current was modeled after Hamada et al. (2003) as a function of the intracellular sodium concentration, IPump = QPump. . 0.62 1+. + 6.7 mM 3 [Na+ ]. i. 0.38 1+. 67.6 mM 3 [Na+ ]i. . ,. (2.6). where QPump was the maximal pump current. While the experimental data in Hamada et al. (2003) corresponds to dorsal root ganglia neurons, the Na+ /K+ pump of cortical neurons of roughly the same size should behave similarly. Indeed, this choice of pump current let to a plausible resting membrane potential and intracellular sodium concentration (Table 2.1).. 2.2.6 KCl Cotransport Under physiological conditions, the chloride Nernst potential is hyperpolarized with respect to the resting membrane potential due to cotransporter-mediated active transport of KCl out of the cell (Blaesse et al., 2009). It is natural to assume that.

(36) 22. 2 A Biophysical Model for Cytotoxic Cell Swelling. the molar cotransporter flux is proportional to the difference of the chloride and potassium Nernst potential (Østby et al., 2009), such that JKCl = UKCl. + [K ]i [Cl- ]i RT ln , F [K+ ]e [Cl- ]e. (2.7). where UKCl was the cotransporter strength, which was chosen to get a resting chloride Nernst potential of approximately ECl- = −80 mV for an extracellular chloride bath concentration of [Cl- ]e = 135 mM.. 2.2.7 Intracellular Concentrations and the Membrane Potential The transmembrane currents and cotransporter flux determined the evolution of . the intracellular molar amounts NX of the different permeant ions X ∈ Na+ , K+ , Cl- , dNNa+ 1 T L = − INa + + INa+ + 3IPump , dt F dNK+ 1 = − IKD+ + IKL+ − 2IPump − JKCl , dt F dNCl1 G L ICl- + ICl − JKCl . = dt F. (2.8). Intracellular concentrations were computed by dividing the molar amounts NX by the intracellular volume W,. [X]i =. NX . W. (2.9). Since we kept track of all intracellular ion amounts, it was not necessary to introduce an additional differential equation for the membrane potential V. It directly followed from the excess of charge and the membrane capacitance C, and was given by V=. F ( NNa+ + NK+ − NCl- − NA- ) , C. (2.10). where NA- was the constant amount of intracellular impermeant anions A- .. 2.2.8 Cell Volume and Water Flux The time course of the cell volume W was determined by the transmembrane water flux JH2 O ,.

(37) 2.2 Materials and Methods. 23. dW = JH2 O . dt. (2.11). Although the exact pathways for the entry of water molecules into neurons are still debated (Andrew et al., 2007), neuronal swelling is driven by an osmotic gradient (Lang et al., 1998). We therefore modeled the transmembrane water flux as JH2 O = LH2 O ∆π,. (2.12). where LH2 O was the effective membrane water permeability and ∆π = RT [S]i − [S]e was the osmotic pressure gradient for ideal solutions (Van’t Hoff, 1887), with [S] denoting the total solute concentration. Finally, we assumed the the total intracellular solute concentration was given by the total intracellular ion concentration,. [S]i = [Na+ ]i + [K+ ]i + [Cl- ]i +. NA. W. (2.13). 2.2.9 Model Parameter Estimation and Validation Recently, neuronal swelling in hippocampal and cortical brain slices of rats was studied by selective modulation of sodium channel kinetics by e.g. veratridine (Rungta et al., 2015). Veratridine blocks the inactivation of the transient sodium current, thereby greatly increasing the membrane sodium permeability (Strichartz et al., 1987). Under these circumstances, the Na+ /K+ pump is no longer able to compensate for the increased sodium influx, and the cell converges to a Gibbs-Donnan-like equilibrium with corresponding changes in cell volume (Figure 2.3A, top trace). Since our model contained a sodium inactivation gate, it was straightforward to perform such veratridine experiments in silico (Figure 2.3B, top trace). This enabled us to estimate the effective cell membrane water permeability, and to validate the model in its prediction of the development of cytotoxic cell swelling by comparing it to experimental data under different conditions (Figure 2.3A, middle and bottom trace). There was excellent agreement with regard to the onset of edema formation, the time course of swelling, and the achieved cell volumes (Figure 2.3B, middle and bottom trace).. 2.2.10 Numerical Implementation All simulations of the model were performed in M ATLAB, using the stiff differential equation solver ode15s..

(38) 24. 2 A Biophysical Model for Cytotoxic Cell Swelling. When the effect of pharmacological blockers was simulated by turning certain currents off or on, they converged exponentially to their new values with a time constant of 30 s. For calculations of the cross section area A we assumed neurons to be spherical, such that A=π. . 3W 4π. 2. 3. .. (2.14). Bifurcation diagrams were created with M ATCONT (Dhooge et al., 2003).. 2.3 Results 2.3.1 Intracellular Osmolarity is Essentially Defined by [Cl- ]i Anions, being negatively charged, express strong forces on cations, and in biological systems the total charge of freely moving cations and anions in a solution is always zero, a condition known as electroneutrality (Nelson, 2003; Plonsey and Barr, 2007). The concentrations as defined in (2.9) include the excess of charge at the cell membrane boundary that generates the membrane potential, and are therefore strictly speaking not equal to the electroneutral bulk concentrations. However, the difference between the two is negligible. For a neuron with a membrane capacitance of C = 20 pF and volume of W = 2000 µm3 , the charge generating a membrane potential of V = ±100 mV corresponds to an intracellular (monovalent) ion concentration of approximately 0.01 mM. This implies that a significant influx of cations (e.g. Na+ ) needs to be either accompanied by efflux of a different cation (e.g. K+ ), netting no change in osmolarity, or by an influx of anions (e.g. Cl- ), increasing the total ion content of the cell. Since the cell membrane is impermeable to the large, negatively charged proteins, chloride is the main permeant anion. Therefore, for a fixed cell volume, the total intracellular ion concentration increases if and only if the intracellular chloride concentration increases.. 2.3.2 Osmotic Pressure in Gibbs-Donnan Equilibrium When all energy-dependent, active transmembrane transport is shut down, a neuron will eventually reach the Gibbs-Donnan equilibrium (Donnan, 1911), a ther-.

(39) 2.3 Results. 25. A. B Veratridine Control (blockers) GlyH-101(50 µM) Low [Cl- ]e (no PTX). 130. h=1. 145. Area (% of baseline). Area (% of baseline). 145. 115. 100. Default parameters G Blockade of ICl Low [Cl- ]e. 130. 115. 100 0. 5. 10. Time (min). 15. 0. 5. 10. Time (min). 15. Figure 2.3: Neuronal swelling after application of veratridine. (A) Experimental data of neuronal swelling in hippocampal and cortical brain slices of rats with mean and SEM, published in Rungta et al. (2015, Figure 3F and Figure 6E). Bath application of veratridine is indicated by a shaded area. Control resembles a blocker cocktail of APV, CNQX, Cd2+ and PTX. Swelling is inhibited by the SLC26A11 blocker GlyH-101 and largely prevented by reducing extracellular chloride to 10.5 mM. Raw data were kindly provided by the Brian MacVicar lab. In the low [Cl]e experiments, the lack of the GABAA Rs blocker PTX leads to an additional chloride influx, which is not taken into account in the model. (B) Model simulations closely mimic experimental results. Application of veratridine is modeled by blocking the sodium inactivation gate. Shown are default parameter values, blockade of the voltage-gated chloride current modeling the effect of GlyH-101, and low extracellular chloride [Cl- ]e = 10.5 mM . Swelling is triggered by a very small and brief excitatory sodium current at t = 2.5 min. For calculation of the cross section area we assume that neurons are spherical. The water permeability of the cell membrane is identical in all three conditions.. modynamic equilibrium which is independent of specific ion permeabilities and in which the Nernst potentials (Nernst, 1888) of all permeant ions are equal to the membrane potential. In our neuron model with three different permeant ion species this implies. [Na+ ]e [ K+ ] e [Cl- ]i , + = + = [Na ]i [K ]i [Cl- ]e. (2.15). where the inverse for chloride results from its valency zCl- = −1. Additionally, the principle of electroneutrality dictates that. [A- ]i + [Cl- ]i = [B+ ]i + [Na+ ]i + [K+ ]i = 21 [S]i , [A- ]e + [Cl- ]e = [B+ ]e + [Na+ ]e + [K+ ]e = 21 [S]e ,. (2.16). where we have added impermeant cations B+ for generality. Combining (2.15) and (2.16), and using the fact that concentrations cannot become negative, yield.

(40) 26. 2 A Biophysical Model for Cytotoxic Cell Swelling. the membrane voltage VGD and solute concentration gradient ∆[S] in Gibbs-Donnan equilibrium. They are given by VGD =. RT ln F. β i − αi +. and ∆ [S] = − αi − β i +. q. respectively, where αi = 12 [S]e − [A- ]i ,. αe = 12 [S]e − [A- ]e ,. q. 2β e β i − αi. αi + β i. 2. 2. ,. (2.17). + 4αe β e. + 4 αe β e − αi β i ,. β i = 21 [S]e − [B+ ]i ,. β e = 21 [S]e − [B+ ]e .. (2.18). (2.19). If we for the moment assume that the cell volume is constant, the osmotic pressure in Gibbs-Donnan equilibrium can be computed with the help of (2.18). For a neuron with a water-permeable cell membrane, convergence to the GibbsDonnan equilibrium is accompanied by an increase in cell volume if and only if ∆[S] > 0, which is equivalent to [ S ] e − 2 [ A- ] e [ S ] e − 2 [ B + ] e αe β e > 1. θ= = αi β i [S]e − 2[A- ]i [S]e − 2[B+ ]i. (2.20). Hence, if the concentration of impermeant cations is equal on both sides of the membrane, cell swelling will only occur if [A- ]i > [A- ]e . It is also apparent that the numerator in (2.20), and therefore the value of θ and the amount of swelling, decreases if we increase the concentration of extracellular impermeant ions, [A- ]e and or [B+ ]e . Quantitative examples are shown for three different extracellular bath solutions (Figure 2.4), with corresponding θ values of approximately 18.5, 1.6, and 3.5. An extracellular solution with physiological concentration of sodium, potassium, chloride and impermeable anions results in a total ion concentration gradient of approximately 160 mM, and Gibbs-Donnan potential of approximately −10 mV (Figure 2.4A). Partial iso-osmotic replacement of extracellular chloride and sodium with impermeant anions and cations, respectively, leads to a significant reduction of the osmotic pressure in Gibbs-Donnan equilibrium (Figure 2.4B and 2.4C)..

(41) 2.3 Results. 27. A. [Na+ ]e = 152.0 mM. [Na+ ]i = 231.9 mM. [ K+ ] e =. [ K+ ] i. 3.0 mM. -. [Cl ]e = 135.0 mM -. [A ]e. = 20.0 mM. =. 4.6 mM. [Cl- ]i = 88.5 mM -. [A ]i. VGD = −11.3 mV. ∆[S] = 163.0 mM. = 148.0 mM. B. [Na+ ]e = 152.0 mM +. [K ]e =. 3.0 mM. -. [Cl ]e = 10.5 mM -. [A ]e. = 144.5 mM. [Na+ ]i = 155.2 mM [ K+ ] i. =. 3.1 mM. -. [Cl ]i = 10.3 mM -. [A ]i. VGD = −0.6 mV ∆ [S] =. 6.6 mM. = 148.0 mM. C [Na+ ]e = 26.0 mM [ K+ ] e =. 3.0 mM. [Cl- ]e = 135.0 mM [A- ]e. = 20.0 mM. [B+ ]e. = 126.0 mM. [Na+ ]i = 153.2 mM [ K+ ] i. = 17.7 mM. [Cl- ]i = 22.9 mM -. [A ]i. VGD = −47.4 mV ∆ [S] =. 31.8 mM. = 148.0 mM. Figure 2.4: Illustration of the Gibbs-Donnan equilibrium for different bath solutions. (A) For a bath solution resembling brain interstitial fluid under physiological conditions, the GibbsDonnan equilibrium is associated with a Gibbs-Donnan potential of VGD = −11.3 mV and a large total solute concentration gradient of ∆[S] = [S]i − [S]e = 163.0 mM. (B and C) Iso-osmotic replacement of extracellular chloride and sodium by cell membrane-impermeant anions A- and cations B+ , respectively, leads to a significant reduction of the total ion concentration gradient ∆[S], and therefore osmotic pressure, in Gibbs-Donnan equilibrium. Qualitative estimates of associated changes in cell volume are indicated with a dashed line. Note that in all situations electroneutrality is preserved and that [Cl- ]i defines the osmotic pressure..

(42) 28. 2 A Biophysical Model for Cytotoxic Cell Swelling. 2.3.3 Ion Permeabilities Determine Speed of Neuronal Swelling Although the Gibbs-Donnan equilibrium and associated Gibbs-Donnan potential do not depend on the (relative) permeabilities of the permeant ion species, ion permeabilities do affect transient behavior and thus determine the time course of reaching Gibbs-Donnan equilibrium and subsequent cell swelling. If water can enter the cell, the Gibbs-Donnan equilibrium becomes dynamic itself, since influx of water will dilute the intracellular concentration of impermeant ions, therefore changing the corresponding equilibrium. Convergence of a neuron from physiological resting state to Gibbs-Donnan equilibrium was simulated by shutting down the Na+ /K+ pump current. Soon after the Na+ /K+ -ATPase was blocked, the membrane potential rose and reached the spiking threshold, which lead to a burst of action potentials that terminated in depolarization block (Figure 2.5A). The cell volume increased to 95% of its final size after approximately 24 h of Na+ /K+ -ATPase blockade (Figure 2.5B). To investigate the role of ion permeabilities in neuronal swelling we simulated the effect of two different channel blockers, which, as expected, did not change the equilibrium volume (Figure 2.5B). However, blockade of the transient sodium current, simulating the effect of TTX, and blockade of the voltage-gated chloride current, simulating the effect of GlyH-101 or DIDS, both slowed down neuronal swelling (Figures 2.5B and 2.5C). In all conditions, the vast majority of swelling resolved after the cell membrane had depolarized, along the branch of Gibbs-Donnan equilibria (Figure 2.5D). Note that the cell volume in the model can increase without bound. In reality, neurons will lyse before their cross section area increases to over 350% of its physiological value.. 2.3.4 Cell Volume Critically Depends on Remaining Pump Activity Thus far we only discussed and simulated conditions with no activity of the Na+ /K+ -ATPase, corresponding to complete anoxia. Our model enabled us to also study compromised pump function, e.g. seen in the penumbral region of patients with ischemic stroke (Liang et al., 2007). Systematic, mathematical study of the dependence of a model on a certain parameter, in our case the strength of the Na+ /K+ -ATPase, is known as bifurcation theory. It permits us to follow equilibria of the model and to detect tipping points, bifurcations, at which the qualitative behavior of the dynamical system changes (Kuznetsov, 2004). Following the physiological resting state while slowly decreasing the Na+ /K+ pump strength revealed that a tipping point exists at approximately 65% of the.

(43) 2.3 Results. 29. A. B IPump = 0. V ENa+ EK+ ECl-. -90 220. 0. 10. Time (min). 300. 200. Default parameters T Blockade of INa + G Blockade of ICl -. 100 -60. 20. C. 0. 24. Time (h). 48. D IPump = 0. 220. Area (% of baseline). Area (% of baseline). 10. -40. IPump = 0. 400. Osmotic equilibrium 0. 180. 140. Default parameters T Blockade of INa + G Blockade of ICl -. 100 -90. 0. 30. Time (min). 60. Potential (mV). Potential (mV). 60. -20 VGD Default parameters T Blockade of INa +. -40. G Blockade of ICl -. 220 -60. Physiological resting state 50. 150. 250. Area (% of baseline). 350. 90. Figure 2.5: Convergence towards Gibbs-Donnan equilibrium and subsequent cell swelling after blocking the Na+ /K+ -ATPase, simulating ouabain perfusion or oxygen-glucose deprivation. (A) Membrane depolarization and evolution of Nernst potentials for default parameters. After blocking the Na+ /K+ -ATPase, the neuron starts spiking for approximately one minute (illustrated by a filled black region), terminating in depolarization block. (B) Time course of the increase in cell volume using blockers for the transient sodium current (simulating the effect of TTX) or voltage-gated chloride current (simulating the effect of GlyH-101 or DIDS). In both conditions, neuronal swelling is slowed down, but the final cell volume is not affected. (C) Close-up of the volume dynamics during the first 90 min after shutdown of the Na+ /K+ -ATPase. Blockade of the transient sodium current prevents spiking and slows down the depolarization of the cell, which yields a delay in the opening of the voltage-gated chloride channel. Blockade of the voltage-gated chloride current limits the chloride flux and therefore the water flux into the cell. (D) Convergence from the physiological resting state towards the osmotically balanced Gibbs-Donnan equilibrium (both denoted by marks). Fast voltage fluctuations due to spiking are averaged out. While converging towards the branch of Gibbs-Donnan equilibria (solid black line), swelling speeds up once the voltage-gated chloride current gets activated (Figure 2.2)..

(44) 30. 2 A Biophysical Model for Cytotoxic Cell Swelling. default pump strength, after which the physiological state disappeared. For pump rates below this critical level, the cell evolved towards a depolarized pathological equilibrium state (Figure 2.6A). At this point, the cell size critically depended on the remaining pump activity. Minor differences in remaining pump strength resulted in major differences in the observed swelling (Figure 2.6B). Vanishing of a stable equilibrium due to collision with an unstable equilibrium is called a saddle-node bifurcation. Close to a saddle-node bifurcation, small changes in circumstances can lead to sudden and dramatic shifts in observed behavior.. 2.3.5 Functionality is not Restored at Physiological Pump Strengths The depolarized, pathological equilibrium corresponds to a Gibbs-Donnan-like state in which the potential energy that is normally stored in the electrochemical ion gradients has largely dissipated (Dreier et al., 2013) and is therefore also known as a state of ‘free energy-starvation’ (Hübel et al., 2014). This equilibrium state appeared stable up to a pump strength of approximately 185% of the default value, such that the model is bistable for a wide range of Na+ /K+ pump rates (Figure 2.6). The model. A. B 370. Area (% of baseline). Potential (mV). 0 Pathological Equilibrium -25 H -50 SN -75 200. Physiological Equilibrium. 150. 100. 280. 190. 100 50. Pump strength (% of baseline). 0. 200. Pathological Equilibrium H Physiological Equilibrium 150. 100. SN 50. Pump strength (% of baseline). 0. Figure 2.6: Bifurcation diagram with the Na+ /K+ -ATPase strength as free parameter. (A) Stable equilibria are denoted by a solid line, unstable equilibria by a dotted line. At approximately 65% of the baseline pump strength, the physiological resting state disappears via a saddle-node bifurcation (SN, orange). For lower values of the pump strength, the cell will converge to a depolarized, Gibbs-Donnan-like equilibrium. This pathological state is stable for pump strengths of up to approximately 185% of the baseline pump rate, where it loses stability due to a subcritical Hopf bifurcation (H, blue). (B) The cell volume is almost constant in the physiological equilibrium branch, but highly dependent on the pump strength in the pathological equilibrium branch, where minor differences in remaining pump rate cause major differences in equilibrium cell size..

(45) 2.3 Results. 31. also predicted that the cell volume may be returned to values near baseline, while the cell membrane is still depolarized if the pump has returned to its baseline value: at a pump strength of 100%, cell volume in the pathological state is approximately 115%, while the membrane voltage is around −35 mV (Figure 2.6). This potential lies within the range where the transient sodium current is partially activated but inactivation is yet incomplete, generating a ‘window’ current (Attwell et al., 1979) that the Na+ /K+ ATPase can not overcome. This implies that, once the cell has converged to this pathological state due to a failure of the Na+ /K+ pump to maintain physiological homeostasis, a return to the physiological resting state is possible only if the pump strength is increased far beyond its nominal value. The loss of stability of the pathological equilibrium is due to a subcritical Hopf bifurcation, at which an unstable limit cycle branches from the equilibrium state. After a dynamical system passes a subcritical Hopf bifurcation point, it will jump to a distant attractor, which, similar to the saddle-node case, can cause dramatic shifts in observed behavior.. 2.3.6 Na+ Channel Blockers May Reverse Cytotoxic Edema If the cell has entered the pathological equilibrium with the associated increase in volume (Figure 2.6), and pump strength returns to baseline or beyond, this state remains. Due to the aforementioned ‘window’ current, the membrane is more permeable to sodium if the cell is partially depolarized. In this state, the sodium current may be too large to be compensated by the Na+ /K+ -pump. Blockade of the voltage-gated sodium current should therefore facilitate a return to the physiological resting state, as it reduces the sodium influx in depolarized conditions. To test this hypothesis, we followed the earlier detected tipping points while slowly reducing the voltage-gated sodium permeability (Figure 2.7A). Indeed, the range of pump strengths that permitted a state of free energy-starvation shrank with decreasing permeability. When the sodium permeability was reduced to less than approximately 40% of its baseline and the pump strength was set to its nominal value, the physiological resting state was the only stable equilibrium. These findings predict that, as long as the Na+ /K+ -ATPase strength is sufficient, temporary (partial) blockade of sodium channels allows cells to return to their physiological resting equilibrium (Figure 2.7B)..

(46) 32. 2 A Biophysical Model for Cytotoxic Cell Swelling. B Physiological Equilibrium. 175. H. -60 125 Bistability 75. SN. Pathological Equilibrium. 25 0. 25. 50. 75. Pump strength (% of baseline). Pump strength (% of baseline). A. H. 72 -60 ZH. 64. GH SN. 56. 48. 100. CP 6. Gated Na+ permeability (% of baseline). 10. 14. 18. 22. Gated Na+ permeability (% of baseline). Area (% of baseline). Potential (mV). C T INa + = 0. IPump = 0. 10. -50. 130. 100 175. 0. 10. 20. 30. 40. Time (min). 50. 60. 70. 80. Figure 2.7: Bistability for physiological pump strengths. (A) Continuation of the saddle-node (orange) and Hopf (blue) bifurcation (denoted by the two marks) with the maximal transient sodium permeability as an additional free parameter. Bifurcation of stable equilibria are denoted by a solid line, bifurcations of unstable equilibria are denoted by a dotted line and shown for completeness. The region of bistability between the solid lines shrinks with decreasing sodium permeability. (B) Magnification with codimension-two bifurcations. The two branches of saddle-node bifurcations meet in a cusp singularity (CP). To the left of this point, the model transitions smoothly between the physiological and pathological state. The Hopf bifurcation curve intersects a saddle-node branch at a zero-Hopf point (ZH) and undergoes a generalized Hopf bifurcation (GH), becoming supercritical. (C) Model simulation illustrating bistability and a possible way to return to the physiological resting state. Transition to the pathological equilibrium is induced by 10 min blockade of the Na+ /K+ -ATPase (simulating ouabain perfusion or oxygen-glucose deprivation). After temporary blocking of the voltage-gated sodium channels (simulating the effect of e.g. TTX) the neuron returns to its physiological equilibrium..

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