INVERSE PROBLEMS FOR THE HODGKIN-HUXLEY MODEL

Proceedings of the Czech Japanese Seminar in Applied Mathematics 2005 Kuju Training Center, Oita, Japan, September 15-18, 2005 pp. 88 94 INVERSE PROBLEMS FOR THE HODGKIN-HUXLEY MODEL MASAYOSHI KUBO 1 Abstract. Inverse problems for the Hodgkin-Huxley equation are examined. To understand functions of a single neuron, such as propagation and generation of synaptic or action potentials, a detailed description of the kinetics of the underlying ionic conductance is essential. In neocortical neurons, network activity can activate a large number of synaptic inputs, resulting in highly irregular subthreshold membrane potential fluctuations, commonly called synaptic noise. This activity contains information about the underlying network dynamics, but it is not easy to extract network properties from such complex and irregular activity. Here we propose a method to estimate properties of network activity from intracellular recordings provides estimates of Na, K and synaptic conductances. This approach should be applicable to in vivo intracellular recordings during different voltage of membrane potential, providing a characterization of the properties of ionic conductances and possible insight into the underlying network mechanism. Key words. Inverse problem, Synaptic noise, Hodgkin-Huxley equation, In vivo intracellular recording, Synaptic conductance AMS subject classifications. 35R25, 35R30, 47A52 1. Introduction. At the present time, there are two broad frameworks of the physiological experimental methods for the information processing of the nerve cells in the neuroscience. On one hand it applies to a small part of in vitro neural organization which is extracted from brain. Another framework is directly experiments on the in vivo nerve cells. The in vitro experiments is relatively easy to apply, and various techniques have already been established in vitro experiments. In vitro physiological experiment is intended only for the minuscule part which is cut off the brain. There is no external input to individual nerve cells. The in vitro nerve cells is alive but seldom or never active [2]. In other words the in vitro nerve cells does not process any information. In the case of in vivo experiments especially by using intracellular recording, we can measure the individual neural activity during information processing. It is difficult to get the in vivo experimental data because of being careful with a living animal. Even if the in vivo activity data is obtained, the analysis and the interpretation of the data are moreover difficult, because the activity of the nerve cells is very complicated: Borg-Graham et.al. [1]. Under such circumstances, it is necessary to establish some analyzing technique for the complicated data of in vivo neuronal activity. The vast majority of the nerve cells generate a series of brief voltage pulses in response to vigorous input. These pulses, also referred to as action potential or spikes, originate at or close to the cell body, and propagate down the axon at constant velocity and amplitude. Various activities of nerve cells were analyzed from the physiological in vivo and in vitro experimental data. Approximately 10 10 nerve cells exist in the cerebral cortex, and there are several thousand of synaptic connections per neuron. In awakening state, the nerve cells of the cerebral cortex incessantly received the external complicated large input from the other connected neurons [2]. The fol- 1 Department of Applied Analysis and Complex Dynamical Systems Graduate School of Informatics, Kyoto University, Kyoto, Japan 88

Inverse Problems 89 lowing is fundamentally important to investigate information processing of the nerve cells; (1) mathematical formulation of membrane potential fluctuations, (2) precise measurement of the physiological experimental data of output from a nerve cell, (3) extracting the external input from the physiological experimental data. Each nerve cell can be regarded as the input/output system. The input is an electrical current which is induced from an external stimulus, and the output is the membrane potential fluctuations and action potentials. It is necessary to make use of the input/output data of a nerve cell to reveal the information processing mechanism of each nerve cell. At the awake state, the membrane potential fluctuations of the nerve cells are highly irregular and their random nature by reason of the external complicated input which is often called synaptic noise. On this account it is difficult to separate the membrane potential fluctuations caused by the external input from the membrane potential fluctuations caused by the current through voltage-dependent ionic channels on the cell membrane. Namely it is inferable that extracting the external input from the physiological experimental data is intractable. From the mathematical point of view it is difficult to obtain the coefficients of the Hodgkin-Huxley equation which represent the membrane potential dynamics of the individual neuron. Depending upon this crucial obstacle there are many open problems related with the information processing of the nerve cells [3],[12],[13],[14]. Here we propose a method in the next sections to estimate properties of neural activity from intracellular recordings provides estimates of Na, K and synaptic input conductances. This approach is applicable to in vivo intracellular recordings during different voltage of membrane potential, providing a characterization of the properties of ionic conductances and possible insight into the underlying network mechanism. 2. The Hodgkin-Huxley Model and the Ionic Channel Conductance. Hodgkin and Huxley [5] carried out a large number of experiments, which lead them to postulate the following phenomenological model of the events underlying the generation of the action potential. They found three different types of ion current, viz., sodium, potassium, and a leak current that consists mainly of Cl ions. Specific voltage-dependent ion channels, one for sodium and another one for potassium, control the flow of those ions through the cell membrane. The leak current takes care of other channel types which are not described explicitly. Hodgkin and Huxley formulated a single equation for the three currents flowing across a patch of membrane, C m dv (t) = I Na (t) + I K (t) + I L (t) + I S (t) + I(t) where V (t) is the membrane potential, I(t) is the current that is injected via an intracellular electrode, and C m is the membrane capacitance. I Na (t) is a sodium current, I K (t) is potassium current, I L (t) is a leak current, and I S (t) is a synaptic external input current. The individual ionic current I X (t) is linearly related to the driving potential via Ohm s low, I Na (t) = G Na m(t) 3 h(t)(v Na V (t)), I K (t) = G K n(t) 4 (V K V (t)), I L (t) = G L (V L V (t)), I S (t) = G Exc (t)(v Exc V (t)) + G Inh (t)(v Inh V (t)) where each of the sodium and potassium ionic conductances is expressed as a maximum conductance, G Na and G K, multiplied by additional variables m, h, and n

90 M. Kubo representing the fraction of the maximum conductance actually open. In other words, the probability that a channel is open is described by m, h, and n. The action potential involves two major voltage-dependent ionic conductances, the sodium conductance and the potassium conductance. They are independent of each other. The third, smaller so-called leak conductance G L does not depend on the membrane potential. G Exc (t) is a synaptic conductance for an excitatory synaptic input and G Inh (t) is a synaptic conductance for an inhibitory synaptic input. The parameters V Na, V K, V L, V Exc,and V Inh are the reversal potentials. Generally speaking, reversal potentials and maximum conductances are empirical given parameters. The synaptic conductances G Exc (t) and G Inh (t) evolve according to the stochastic differential equations, dg Exc (t) = 1 (G 0 Exc τ G 2σExc 2 Exc(t)) + dbt Exc, Exc τ Exc dg Inh (t) = 1 (G 0 Inh τ G 2σInh 2 Inh(t)) + dbt Inh Inh where τ Exc and are time constants for the excitatory synaptic input and the inhibitory synaptic input, respectively. G 0 Exc and G0 Inh are the averages of the excitatory and inhibitory synaptic input conductance, and σ Exc and σ Inh are the standard deviations of the excitatory and inhibitory synaptic input conductance, respectively. Bt Exc and Bt Inh are independent Brownian motions. The three variables m(t), h(t), and n(t) are called activation parameters or gating variables. They describe the state of fictional activation particle and are dimensionless number between 0 and 1. They evolve according to first-order differential equations, dm(t) = α m (V (t))(1 m(t)) β m (V (t))m(t), dh(t) = α h (V (t))(1 h(t)) β h (V (t))h(t), dn(t) = α n (V (t))(1 n(t)) β n (V (t))n(t) where α m (V ), α h (V ), α n (V ) are voltage-dependent rate constant, specifying how many transitions occur between the closed and the open states, and β m (V ), β h (V ), β n (V ) express the number of transition from the open to the closed states. The various functions α X (V ) and β X (V ) are empirical functions of the membrane potential V that have been adjusted by Hodgkin and Huxley to fit the data of neurons [5],[6]. 3. Formulation and Analysis of the Inverse problems. In this section we propose a mathematical method to estimate the amount of the ionic channels for each neuron by using the in vivo membrane potential data. The coefficients is directly related to the total ionic channel s conductance. The maximum conductance for Na channels G Na is equivalent to the product of the amount of the Na channels and the conductance for one Na channel. G Na = (the amount of the Na channels) (the conductance for one Na channel). We consider a problem to find coefficients G Na, G K, G L of the Hodgkin-Huxley (H- H) equation and external synaptic input by using the given data V (t) and I(t). This

Inverse Problems 91 problem is a nonlinear inverse problem and in the case of the external complicated input the problem is difficult to estimate the coefficients [10]. We consider an inverse problem to find the maximum conductances and synaptic input from the intracellular recording in vivo. We assume that the membrane potential V (t) and the current I(t) satisfy the following equations, C m dv (t) = G Na m(t) 3 h(t)(v Na V (t)) + G K n(t) 4 (V K V (t)) + G L (V L V (t)) + I S (V (t), t) + I(t), I S (V (t), t) = G Exc (t)(v Exc V (t)) + G Inh (t)(v Inh V (t)), dm(t) = α m (V )(1 m) β m (V )m, dh(t) = α h (V )(1 h) β h (V )h, dn(t) = α n (V )(1 n) β n (V )n, dg Exc (t) = 1 τ Exc (G 0 Exc G Exc (t)) + dg Inh (t) = 1 (G 0 Inh G Inh (t)) + 2σ 2 Exc db t, τ Exc 2σInh 2 db t where C m, V Na, V K, V Exc, V Inh, and α X (V ), β X (V ) (X = m, h, n) are fixed given parameters and functions. V (t) and I(t) are observation data. Our inverse problem is to find the maximum conductances G Na, G K, G L, G 0 Exc, and G0 Inh, the reversal potential V L, and the synaptic input I S (V, t). Here τ Exc,, σ Exc, and σ Inh are unknown parameters. A profound advance in electrophysiology came with the development of the gigaseal and patch-clamp methods. Erwin Neher and Bert Sakmann wanted to record from a tiny area ( patch ) of surface membrane by pressing a firepolished pipette against a living cell. In 1976 they reported the first single-channel current records with an acetylcholine-activated channel. But the real breakthrough was reported in 1981, when they showed that clean glass pipettes can fuse with clean membranes to form a seal of unexpectedly high resistance and mechanical stability. They called the seal a gigaseal since it could have an electrical resistance as high as tens of gigaohms (giga= 10 9 ). For their discoveries concerning the single ion channels in cells, Erwin Neher and Bert Sakmann shared the Nobel Prize in physiology or Medicine in 1991. In a voltage patch clamp experiment one controls the membrane potential V (t) and measures the transmembrane current I(t) required to maintain that voltage [11]. By using this technique, for each holding potential V (t) = V j, we measure the obser-

92 M. Kubo Relative Error of Na Conductance 1.0 0.8 0.6 Na numerical data linear fitting 0 4.0 8.0 12.0 SD of Synaptic Conductance [n ] Ω Relative Error of K Conductance 1.0 0.8 0.6 0.4 K numerical data linear fitting 0.2 0 4.0 8.0 12.0 SD of Synaptic Conductance [n ] Fig. 3.1. Numerical results for the maximum ion channel conductances; (Left) relative error of G Na, (Right) relative error of G K. Ω vation current I = I j, then V j and I j satisfy the following equations, G Na m (V j ) 3 h (V j )(V Na V j ) + G K n (V j ) 4 (V K V j ) + G L (V L V j ) + I S (V j, t) + I j (t) = 0, I S (V j, t) = G Exc (t)(v Exc V j ) + G Inh (t)(v Inh V j ), dg Exc (t) = 1 (G 0 2σExc 2 Exc G Exc (t)) + db Exc τ Exc τ Exc dg Inh (t) = 1 (G 0 Inh G Inh (t)) + t, 2σInh 2 dbt Inh where m (V ), h (V ), and n (V ) are equilibrium states for the activation parameters: m (V ) = α m (V ) α m (V ) + β m (V ), h (V ) = α h (V ) α h (V ) + β h (V ), n (V ) = α n (V ) α n (V ) + β n (V ). Here our transformed inverse problem is to find the maximum conductances G Na, G K, G L, G 0 Exc, G0 Inh, the reversal potential V L, and the synaptic input I S from the observation data V j, I j and the given data V Na, V K, V Exc, V Inh, m (V ), h (V ), n (V ). G Exc (t) and G Inh (t) describe the Ornstein-Uhlenbeck processes, and it is easily verified by Itô s rule that E[G Exc (t)] = G 0 Exc, E[G Inh(t)] = G 0 Inh where for a random variable X, E[X] describes the expectation of X. We set I 0 j := E[I j(t)] then we obtain the following equation for each holding potential V j (j = 1, 2, ), G Na m (V j ) 3 h (V j )(V Na V j ) + G K n (V j ) 4 (V K V j ) + G L (V L V j ) + G 0 Exc(V Exc V j ) + G 0 Inh(V Inh V j ) + I 0 j = 0. Here the above problem is not nonlinear differential equations but nonlinear equations. The retransformed problem is to find G Na, G K, G L, V L, G 0 Exc, and G0 Inh from the

Synaptic Input Current [na] 1.8 1.4 1.0 0.6 Inverse Problems 93 Exact Current Numerical Data 0.2 0 10 20 30 40 50 time [ms] Fig. 4.1. Comparison of numerical results for external synaptic input current with the exact data. observation data V j, I j and the known data V Na, V K, V Exc, V Inh, m (V ), h (V ), n (V ). In order to solve the problem, it is necessary to use the Newton method. By using the above solution (G Na, G K, G L, V L, G 0 Exc, G0 Inh ), we obtain the synaptic input I S (V j, t) as follows. I S (V j, t) =I j (t) ( G Na m (V j ) 3 h (V j )(V Na V j ) + G K n (V j ) 4 (V K V j ) + G L (V L V j ) ) 4. Numerical Results. In this section we show numerical results for the transformed inverse problems by using the Newton s method with Tikhonov s regularization. All of the results in this section were obtained by a pseudo-random procedure ( Mersenne Twister random number generator [8]). We need to apply Tikhonov s regularization because of the measurement error for observation data and the ill-posedness of this inverse problem. We refer to the physiological experimental given value of cortical pyramidal neuron; V Na, V K, V Exc, V Inh, G 0 Exc, G0 Inh, τ Exc,, σ Exc, σ Inh. Figure 2.1 presents the numerical results for the maximum conductances for sodium channels and potassium channels (G Na, G K ). The relative error of sodium conductance is comparatively small even if the synaptic input is exceedingly large deviation. Figure 2.2 presents the numerical results for synaptic input I S (V j, t). In each case the numerical solution of the inverse problem is obtained exactly and stably by the above technique. 5. Concluding Remarks. The elucidation of information processing in the brain which is always taken up is one of the final aims of neuroscience researches. It should be noted that the neuronal dynamics is related to not only output information but input information in order to investigate information processing of the individual neuron. Here the input is synaptic external input current and the output is the dynamics of membrane potential or action potentials. By the technological advance we can measure the output data of each neuron and there are many kind of useful simulation software to represent the membrane potential dynamics of the individual neuron. But to quantify the input to each neuron is a pending issue although it is exceedingly important to investigate information processing [10]. In the awake state the intracellular recording data shows that the membrane potential randomly and greatly fluctuates. It is confirmed that the membrane potential

94 M. Kubo fluctuation is attributable to the external synaptic input in vivo [7]. The main ingredient in the making of the membrane potential fluctuation is not only the external synaptic input but also the voltage-dependent sodium and potassium channels on the membrane. This is the reason why it is difficult to extract the external synaptic input from the data of the membrane potential in vivo which follows the nonlinear differential equations. At the present stage in the neuroscience the average of the membrane potential is regarded as the approximation of external input because of such difficulty. Through the modern experimental technique, our method in the 3 pave the way for employing the observation data V (t) and I(t) for calculating the coefficients of the Hodgkin-Huxley equation and external current input. In a sense, the coefficients of the Hodgkin-Huxley equation essentially relate the amount of the ion channels on the membrane. Our method represent a step on the way to estimate the quantitative alteration of the amount of ion channels on the membrane for each neuron, and it would be useful to research the relationship between learning mechanism in the brain and the increase and decrease in the distribution of ion channels. REFERENCES [1] L.J. Borg-Graham, C. Monier & Y. Fregnac, Visual input evokes transient and strong shunting inhibition in visual cortical neurons, Nature, 393 (1998), 369 373. [2] A. Destexhe, M. Rudolph & D. Pare, The high-conductance state of neocortical neurons in vivo, Nature Reviews Neuroscience, 4 (2003), 739 751. [3] D. Durstewitz, J.K. Seamans & T.J. Sejnowski, Neurocomputational models of working memory, Nature Neuroscience, 3 (2000), 1184 1191. [4] C.R. Froemke & Y. Dan, Spike-timing-dependent synaptic modification induced by natural spike trains, Nature, 416 (2002), 433 438. [5] A.L. Hodgkin & A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J.Physiol., 117 (1952), 500 544. [6] B. Hille, Ion channels of excitable membranes (3rd. ed.), Sinauer, (2001). [7] B.Q. Mao, F. Hamzei-Sichani, D. Aronov, R.C. Froemke & R. Yuste, Dynamics of spontaneous activity in neocortical slices, Neuron, 32 (2001), 883 898. [8] M. Matsumoto and T. Nishimura, Mersenne Twister: A 623-Dimensionally Equidistributed Uniform Pseudo-Random Number Generator, ACM Transactions on Modeling and Computer Simulation, 8 (1998), 3 30. [9] D. Pare, E. Shink, H. Gaudreau, A. Destexhe & E.J. Lang, Impact of spontaneous synaptic activity on the resting properties of cat neocortical neurons in vivo, J. Neurophysiol., 79 (1998), 1450 1460. [10] M. Rudolph, Z. Piwkowska, M. Badoual, T. Bal & A. Destexhe, A Method to estimate synaptic conductances from membrane potential fluctuations, J. Neurophysiol., 91 (2004), 2884 2896. [11] B. Sakmann & E. Neher, Single-Channel Recording (2nd ed.), Plenum Press (1995). [12] M.N. Shadlen & W.T. Newsome, The variable discharge of cortical neurons: implications for connectivity, computation and information coding, J. Neurosci., 18 (1998), 3870 3896. [13] Y. Shu, A. Hasenstaub & D.A. McCormick, Turning on and off recurrent balanced cortical activity, Nature, 423 (2003), 288 293. [14] W. Softky & C. Koch, The highly irregular firing of cortical cells in inconsistent with temporal integration of random EPSPs, J. Neurosci., 13 (1993), 334 350. [15] M. Steriade, I. Timofeev & F. Grenier, Natural waking and sleep states: a view from inside neocortical neurons, J. Neurophysiol., 85 (2001), 1969 1985.