Analog circuits for modeling biological neural networks - CiteSeerX
Analog circuits for modeling biological neural networks: design and applications.
S. Le Masson, A. La aquière, Student Member IEEE, T. Bal and G. Le Masson S. Le Masson and A. La aquière, Laboratoire de Microélectronique IXL, CNRS UMR 5818, ENSERB, Université Bordeaux I, 351 cours de la Libération, 33405 Talence, France. T. Bal, Institut A. Fessard, CNRS UPR 2212, avenue de la Terrasse, 91198 Gif-surYvette, France. G. Le Masson, Institut de Neurosciences F. Magendie, unité INSERM 378, 1 rue C. Saint-Saëns 33077 Bordeaux, France.
Abstract Computational neuroscience is emerging as a new approach in biological neural networks studies. In an attempt to contribute to this eld, we present here a modeling work based on the implementation of biological neurons using speci c analog integrated circuits. We rst describe the mathematical basis of such models, then present analog emulations of dierent neurons. Each model is compared to its biological real counterpart as well as its numerical computation. Finally, we demonstrate the possible use of these analog models to interact dynamically with real cells through arti cial synapses within hybrid networks. This method is currently used to explore neural networks dynamics.
Key Words Analog circuits design, biological neurons, hybrid neural networks, neural modeling, stomatogastric ganglion, thalamus.
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1 Introduction Biological neural networks are made of complex, highly non-linear elements, connected within networks containing thousands of interconnected neurons. The study of such networks, their properties and dynamic behavior could be limited by classical experimental approaches. One way to overcome these limitations is to study biological networks from a theoretical point-of-view, using mathematically de ned models of the neurons [1]. Various formalisms have been developped, and amongst them, the Hodgkin-Huxley's description of neuronal excitability [2], that has been shown to be one of the most fruitful and powerful framework for realistic neuronal modeling [3]. Computational neuroscience commonly uses it, by digital simulations on microcomputers or workstations. The well-known neurosimulator softwares [4] compute conductancebased models, which is a time-consuming task when the model complexity grows up. Consequently, real time computation is clearly di cult, even with the fastest microprocessors. More recently, were developed electronic analog models able to compute the HodgkinHuxley formalism [5], [6], [7]. They accurately reproduce the electrical properties of real neurons, as well as the time and voltage dependence of the membrane conductances. Moreover this electrical activity is replicated in real time. Computation speed could even be scaled-up by simply changing the time constants. However, all the analog models that have been developed present some level of abstraction, depending on the applications they are built for. We present in this paper the integrated circuits we designed speci cally for the realization of analog models that represent very detailed simulations of neural electrical activity. Indeed, our goal is to use these circuits as a tool for computational neuroscience. These speci c integrated circuits (ASICs) are designed in full custom mode using a bipolar and complementary MOS technology (BiCMOS). We will show how we reproduced the conductance-based structure of the mathematical model, and designed elementary blocks that can be programmed to emulate any conductance by an externally tuned set of parameters. A single chip contains several 2
conductance blocks, that can be independently connected to others to form a complex neuron or network. Finally, we will present the results of experimental work using `hybrid networks', i.e. networks interconnecting in real time modeled neurons and real ones in an biological preparation [8].
2 Realistic conductance-based models The electrical activity of a neuron is the consequence of the diusion of ionic species through its membrane [3]. This activity is characterized by a membrane potential, which is the voltage dierence between the outside and the inside of the cell. Ions ow through the cell membrane by ion-speci c channels, generating ionic currents. To each ion type is associated a reversal potential, due to the dierence between the intracellular and the extracellular concentrations, and following the Nernst equation. Ions tend to diuse through the membrane, attracting the membrane potential towards their respective reversal potential. For each ion type, the fraction of opened channels, which results from the interaction between time and voltage dependent activation and inactivation process, detrmine the global conductance of the membrane to that ion. Realistic conductance-based models of neurons are usually described using the Hodgkin-Huxley based approach [2]. In that formalism, the current owing across a membrane is integrated on the membrane capacitance, following the electrical equation (1), where Vmem is the membrane potential, Cmem the membrane capacitance, and IS an eventual external current (stimulation or synaptic input). A leak current Ileak is also considered, due to a small but permanent permeability of the membrane to some ions. The associated leak conductance gleak is generally non voltage dependent (2). Each ionic current Ii is described by a set of equations and parameters: Ii is the current passing through the ith channel type, given by equation (3) and (4), in which gmaxi is the maximal conductance value; mi and hi are the dynamic functions describing the permeability of membrane channels to this ion, represented by opening (activation mi) and closing (inactivation hi) fractions. As we see in equation (5), mi tends with a 3
time constant mi to its associated steady-state value m1i , which is a sigmoïdal function of Vmem . hi follows the same dynamics, with an inversely sigmoïdal steady-state value h1i . [Table 1 must be here] In ionic currents, additional expressions of interdependencies may in some case be necessary to express complex neural activities. Studies showed that some ionic currents do not only depend on the time and voltage variables, but also on intracellular calcium concentration. In that case, the corresponding activation (or inactivation) porcess is modi ed, as the calcium concentration changes. The calcium concentration is computed by a rst-order dierential equation depending on the ionic calcium current (equations (7) and (8)). All the parameters of the models following this formalism are experimentally determined, generally using voltage-clamp methods. They are then considered as biologically realistic models and usable in hybrid experiments (see section V). Synaptic transmission is another important concept of neural networks, as they are essential for information transmission between the cells. The synaptic current Isyn owing across the synapse is modeled as in equation (9), where Vmempre and Vmempost are respectively the presynaptic and postsynaptic membrane voltages. The synaptic conductance gsyn is described by the same formalism as votage-dependent channels (equations (4) to (6)). In hybrid experiments, one or more living neurons are connected to modeled neurons by arti cial synapses [8] and the real neuron may be either the presynaptic or postsynaptic. Thus we need to measure in real time its membrane voltage (in order to compute synaptic currents for both the living and modeled neurons) and to inject across the membrane a real synaptic current issued from the arti cial neuron. One solution is to use as an interface a specialized ampli er (Axoclamp 1B from Axon Instruments ) in a multiplexing mode (DCC for Discontinuous Current Clamp) allowing voltage measurement and current injection with a single glass intracellular 4
microelectrode impaled into the living neuron. That constraining structure (microelectrode + Axoclamp for each couple of synapses) makes experiments using those synaptic connections quite complex to handle, therefore only two or three couples of arti cial synapses can reasonably be considered in hybrid reconstructions.
3 Circuits implementation The circuits are ASICs (Application Speci c Integrated Circuits) designed with a 1.2 m BiCMOS technology, allowing the implementation on a chip of both bipolar and CMOS transistors. Bipolar transistors are used for the computation of sensitive functions (classically logarithmic, exponential or multiplicative). They also appear in blocks such as current conveyors or mirrors, to improve the precision and/or the output impedance. CMOS transistors help reducing the power consumption and minimize the chip area. Using that technology, we designed elementary modules [6], [7], which function is to behave as ionic current generators : they can be described as voltage-controlled current sources, with a transconductance replicating the Ii = f (Vmem ) function of the HodgkinHuxley formalism (with addition of the calcium-dependence equations). Due to technological constraints, we have a xed gain scaling Ii and Vmem variables. Real ionic currents in a neuron are nanoamperes, and membrane voltages are in the -100 mV to +100 mV range. Voltages on the chips have a gain of 10, and currents a gain of 1000, which implies a gain of 100 on conductance elements. We shall rst describe the implementation of a single module, i.e. an ionic current generator, that covers an area of approximately 1.5 mm x 2 mm. Mathematical functions such as addition, exponential, multiplication or derivation are computed by elementary circuits handling a small number of transistors. We minimized these numbers by designing the circuit in current mode [8]. All internal variables are treated as currents, and we can exploit the basic transfer functions of transistors that result in current outputs. However, this type of analog computation requires a very precise 5
design, as each single transistor in uences directly the transfer function; the circuits also has to be insensitive to the absolute sizing of transistors, which depends on the precision level of the technological implementation. The modules were then integrated on the ASICs in a 'full custom' mode. That mode of design allows an accurate sizing of each transistor, and ensures the relative sizing precision after fabrication. An analog model of a biological neuron is constructed as shown in gure 1, where each ionic current generator is as described above. A multi-conductances neuron is formed by externally connecting the current outputs on a membrane capacitance and a small circuit that computes the leak current (equation (2)). The measured membrane voltage is reinjected in the module as an equation variable. A single chip can actually include 2 or 3 conductances; each one has its own current output and Vmem input. A neuron is constructed by connecting a number of modules, coming from one or more chips. On the other side, modules on a single chip can be shared between dierent neurons, or be dedicated to synaptic connections, as we will see later. [Figure 1 must be here] One important speci city of our circuits is that none of the equations parameters is constraint. To each one is associated a chip input, on which the user applies a voltage that drives an internal voltage or current . These voltages or currents represent the Hodgkin-Huxley equation parameters, and correspond for each ionic current to the following terms: m1i (Vmem ); h1i (Vmem ); mi ; hi ; Vequii , gmaxi . To obtain such programmability, we separated each module is separated in several blocks as follows: 1. A sigmoïd generator is used to emulate the steady-state m1i and h1i functions (equation (7)); This function is de ned by an oset and a slope voltage. The Voffseti parameter is corresponding to the half-activation or half-inactivation potential, and Vslopei determines the slope of the linear part of the sigmoïd. The electronic representation of this function is based on a classical bipolar dierential pair, which current transfer function is a normalized sigmoïdal function. 6
The Voffset and Vslope parameters are introduced in the sigmoïdal function by a dierentiation and a multiplication applied on the input. 2. The output of the sigmoïd generator is connected to the next block, which computes equation (4) using a classical integrating scheme. The kinetic (identi ed with the m or h parameter) is set by a RC product, in which R is an on-chip resistor, and C an external capacitance, connected to a pin of the chip. We chose to leave the capacitance as an external component to be able to reproduce very slow kinetics, appearing in some models: they can go up to a few seconds, and in that case, the capacitance would be too large to be integrated. It should be noted that an approximation remains in the analog implementation where kinetics are not voltage-dependent, in contrast with the original model. We found out that voltage dependency of kinetics can be skipped in most models without degrading the result, if the kinetic value is carefully chosen. This is the case in the applications we present in sections IV and V. However, this non-dependence is a defect that cannot be neglected in all cases. Various implementation solutions are studied to solve that problem. 3. Finally the ionic current is the output of two cascaded multipliers, that generate successively gmaxi mpihqi and gioni (Vmem , Vequii ). These multipliers operate in current mode, and their output are normalized to respect the gain constraints we de ned at the beginning of the section (arti cial neuron currents = 100 times real neuron currents). The typical output range varies from -100 A to +100A. 4. The calcium-dependence equation is implemented in a separate module, designed to process equations (7) and (8), and to be interconnected with the ionic current generators. Its variable inputs are the ICa current, and an activation value m1 of the ionic current chosen as the calcium-dependent one. The output m1(Ca) is driven back to the source generator and input as the steady-state value m1 of equation (5). As before, and C are tunable parameters, using an external capacitance for the time constant. 7
[Figure 2 must be here]
4 Modeling neurons Computational neuroscientists have developed accurate models of neurons or neural networks presenting well-de ned behaviors. Model neural networks can faithfully reproduce the global activity resulting from the interaction of neurons within a real network, even though the simulated electrical activity of individual modeled neurons does not strictly t the electrical membrane activity of real neurons. This explains the slight dierences in electrical activity that may appear between real neurons and models. The results we present here only intend to prove the correct replication of the numerical model by an analog one. In fact, neuroscientists construct and validate models by determining the role of each parameter in shaping the individual cell response to synaptic or external stimulations, and by looking at the network overall behavior. To validate our own analog models, we directly compare the membrane potential coming from three sources: The experimental measurement on a living cell, through a microelectrode impaled into the neuron. The numerical computation of a mathematical model following the HodgkinHuxley formalism; this computation runs on a dedicated simulation software (`Maxim' software, by G. Le Masson; `Maxim' has been developed in C and runs on Apple Macintosh computers). The analog computation of the same model, using the ASICs. To further demonstrate the validity of our analog models, we present in the following paragraph some results of experiments where we con gured identical model chips with dierent parameters taken from diernet types of neurons. In each case, the membrane voltages measured from the analog model are compared with other voltage representations. The chips parameters are then the ones of the mathematical model, 8
only modi ed by the scale factors on currents and voltages indicated in section III. 4.1
Invertebrate neurons
The two-conductances model originally de ned by Hodgkin-Huxley runs spontaneous action potentials (spikes), using an inward Sodium current, with both activation and inactivation processes, and an outward Potassium current. However, most real neural cells present a more complex activity than simple spikes. In invertebrate species, a well-studied neural network is the lobster stomatogastric ganglion (STG) [9], [10]; one of the well identi ed cell of this network is called Pyloric Dilator (PD) and is modeled by a set of 7 ionic conductances, one of which is calcium-dependent. The resulting spontaneous activity is a succession of bursts of action potentials on top of a slowly oscillating envelop. This activity is quite sensitive to the precision of the equation computation and to the accuracy of the parameters, that have been experimentally de ned [11]: the equations have to be very well emulated by the ASICs to result in a correct membrane voltage. As previously mentionned, the only dierence with the original model is the absence of the voltage-dependence in the time constants. Figure (3 A) is a recording of a typical spontaneous bursting activity in a real PD cell. Comparing this reference activity to both the numerical model (3 B) and the integrated analog equivalent (3 C), reveals a good match of the overall shape and frequency of the signals. [Figure 3 must be here] 4.2
Vertebrate neurons
We also got interested in modeling neurons from vertebrates. We chose neurons from the thalamus for several reasons including the available accurate description of their ionic conductances, and the relative simplicity of the thalamic network. The thalamus is a grey matter nucleus that relays to the cortex most of the sensory information during awake state (including visual information), and becomes the generator of some types 9
of rythms (spindle waves) during early stages of sleep. Inside this structure, synaptic interactions between two populations of thalamocortical neurons are responsible for the generation of spindle waves that waxes and wanes periodically in the thalamocortical networks [12], [15]. The ability of that intra-thalamic network to produce a coordinated rhythmic activity could be a fundamental property involved in the gating of sensory information to the cortex and might be involved into paroxysmal synchronized oscillations resembling those underlying generalized absence seizures. We implemented a model of a neuron involved in such an intra-thalamic circuit, a nucleus Reticularis neuron, or nRt cell [13]. This model includes ve conductances, one of which is calcium-dependent. These conductances give the cell an endogeneous bursting behavior, occurring in response to intracellular negative current injection (4 A). Our analog model of a nRt neuron expresses the same behavior (4 B). [Figure 4 must be here]
5 Networks and hybrid networks The next step of the circuits evaluation process is to look at the emulation of neural networks, where neurons activity is driven by synaptic interactions. In that case, we use arti cial synapses implemented as indicated in section II, for the construction of hybrid networks [14]. In those systems, modeled neurons interact in real time with in vitro living cells; the hybrid system method gives the opportunity to access and tune separately the parameters of endogenous or synaptic conductances, and then to infer their individual role in shaping the entire network behavior. Modeled conductances or neurons are either added to the initial network, or replace a previously inactivated cell. Hybrid systems have been initially implemented using numerically-computed models, where the model computation speed is directly proportional to the number of conductances taken into account. The principle of the hybrid systems implies that the model runs in real time; one understands easily that limitations, due to the numerical computation time, quickly appear when the model complexity grows up. Analog com10
putation de nitely solves the problem of real time, provided that time constants tuned on the chips are identical to those of the real neurons. As for the isolated modeled cells, we worked with hybrid systems on vertebrate and invertebrate neural networks. Hybrid networks were constructed using the models described in section IV (PD and nRt neurons), as these models have been perfectly characterized and validated [13], [14]. 5.1
Invertebrate neural network
We will consider at rst an hybrid system, which wiring diagram is shown in 5 A. A PD model is connected to a spontaneously bursting neuron, recorded from an invertebrate stomatogastric ganglion. All synapses are inhibitory. Figure 5 B to 5 D show the eect of a linear increase of the calcium current in the model (by the mean of an increase of the gmaxCa parameter). Without calcium current, the model behaves as a tonic cell and res action potentials. When connected with reciprocally inhibiting synapses, the two cells organize a diphasic rhythm of alternate bursts (5 B). If the calcium maximal conductance is increased, the model expresses a more complex response to the real neuron's synaptic inhibition: for an intermediate value of gmaxCa , the model neuron responds by a single burst followed by a tonic phase (5 C), whereas for stronger values, it has a full bursting activity (5 D). In this latter case, the network is stable with a 3 to 1 entrainment. In all cases, one can note the synaptic hyperpolarization in the real neuron, due to the in uence of bursts and action potentials in the model neuron. [Figure 5 must be here] 5.2
Vertebrate neural network
Our method is also particularly well suited for constructing hybrid networks using vertebrate brain slices. In such preparations, due to slice thickness (400 m), long range synaptic connections are often cut, except for very local ones. In the visual thalamus, 11
relay cells (also named TC for thalamocortical) directly receive inputs from the retina. Their connections in a feed forward way to the visual cortex allow the visual inputs to be transmitted to the dedicated treatment areas. But within the thalamus, the TC cells are also connected to another thalamic nucleus (nucleus Reticularis), and its inhibitory nRt cells [15]. The global synaptic diagram between TC and nRt cells is a simple excitatory (from TC to nRt) and inhibitory (from nRt to TC) loop, as shown in 6 A. Depending on the state of this neuronal loop, visual ow can be accurately transmitted to the cortex (awake state), or actively ltered by strong oscillations (sleep-like state). These oscillations are the result of the synaptic interactions between the neurons, but also of their intrinsic cellular properties. In only one specie, the ferret, synaptic connections between TC and nRt cells are well preserved after slicing. Working on other species (rats or guinea pigs), and using the hybrid networks method with an analog model of a nRt cell, we were able to reconstruct a functional circuit that generates sleep-like oscillations (spindle waves). (6 B) and (6 C) show the resulting activity of our arti cial network, in response to a short negative current injection in the TC neuron (time t1 ). This stimulation triggers a long lasting (0.5 s to 1 s) oscillation of the membrane potential in both cells. The oscillation sequence is based on the following cellular and synaptic mechanisms: the negative current step injection induces a rebound calcium potential and a burst of spikes in the TC cell resulting in the synaptic excitation of the modeled nRt cell. In turn, the ring of the nRt cell results in strong inhibitions of the TC cell leading to more rebound calcium potentials. Due to that handshake sequence, the system nally oscillates at a 12 to 14 Hz frequency, until a slow calcium-dependent current terminates the wave (time t2). This simple experiment illustrate how hybrid systems can facilitate the analysis of network activity in experimental situations where classical approaches are not possible. [Figure 6 must be here]
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6 Conclusion We have presented in this paper a summary of the design principles and applications of a set of analog integrated circuits that we developed to emulate mathematical models of biological neurons. That type of circuits is an interesting evolution in computational neuroscience, due to the following characteristics: - on-chip programmability of the model parameters, in order to use identical circuit in dierent con gurations to emulate dierent neurons. - better circuit integration (i.e. smaller die size) than an equivalent digital implementation. - intrinsic real time behavior, as circuits are an electronic equivalent of the mathematical description (including the time constants). That latest property is exploited in the hybrid networks experiments, where neural networks are reconstructed using both arti cial and living neurons. For a purely computational application, including for example regulation functions (that are very important for the evaluation of functional stability in neural networks), it is very easy to accelerate the computation speed of the electronic models, only by multiplying all the time constants by an identical factor. Cut-o frequencies of the used BiCMOS technology for our type of design is high enough (around 50 MHz) compared to the original model one (a few kHz) to allow a computing scaling-up by a factor of at least 1000. In comparison, the numerical computation of the electrical activity of a nRt cell modeled as in section IV B could not be done with a sampling rate better than 10 kHz, using an optimized version of the neurosimulator software 'Maxim' running on a 132 MHz Apple computer (Power Macintosh 9500).
REFERENCES [1] B. Softkey, C. Koch, Single cell models, in M. Arbib, editor, The handbook of brain theory and neural networks, pp. 879-884, MIT Press, Boston, MA, 1995. [2] A.L. Hodgkin and A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, Journal of Physiology, vol. 117, pp. 500-544, 1952. [3] C. Koch and I. Segev, Editors, Methods 13 in neuronal modeling: from synapses to
networks, MIT Press, Cambridge, MA, 1989. [4] J. Murre, Neurosimulators, in M. Arbib, editor, The handbook of brain theory and neural networks, pp. 634-639, MIT Press, Boston, MA, 1995. [5] C. A. Mead, Analog VLSI and neural systems, Addison Wesley, Reading, MA, 1989. [6] R. Douglas and M. Mahowald, A construction set for silicon neurons, in Neural and Electronics Networks, F. Zornetzer, Editor, Academic Press, Arlington, 1995. [7] E. A. Vittoz and X. Arreguit, Systems based on bio-inspired analog VLSI, MicroNeuro'96 tutorial, Lausanne, Feb. 12-14, 1996. [8] G. Le Masson, S. Le Masson and M. Moulins, From conductances to neural networks properties: analysis of simple circuits using the hybrid networks method, Progress in Biophysics and Molecular Biology, vol.64 n 2/3, pp. 201-220, 1995. [9] D. Dupeyron, S. Le Masson, Y. Deval, G. Le Masson and J.P. Dom, A BiCMOS implementation of the Hodgkin-Huxley formalism, Proc. of MicroNeuro'96, Lausanne, IEEE Computer Society Press, pp. 311-316, 1996. [10] A. La aquière, S. Le Masson, D. Dupeyron and G. Le Masson, Analog circuits emulating biological neurons in real-time experiments, IEEE International Conference on Biomedical Engineering (EMBS'97, Chicago, IL), pp. 2035-2038, October 1997. [11] C. Toumazou, F.G. Lidgey and D.G. Haigh, Analog IC design: a current-mode approach, IEE circuits and systems, London, 1990. [12] J. Golowasch, F. Buchholtz, IR. Epstein, E. Marder, Contribution of individual ionic currents to activity of a model stomatogastric ganglion neuron, Journal of Neurophysiology, vol. 67, pp. 341-349, 1992. [13] R.M. Harris-Warrick, E. Marder, A.I. Selverston and M. Moulins, editors, Dynamic biological networks: the stomatogastric nervous system, Cambridge, MA, MIT Press, 1992. [14] G. Le Masson, E. Marder and L.F. Abbott, Activity-dependent regulation of conductances in model neurons, Science, vol. 259, pp. 1915-1917, 1993. [15] M. von Krosigk, T. Bal, D.A. McCormick, Cellular mechanisms of a synchro14
nized oscillation in the thalamus, Science, vol. 261, pp. 361-364, 1993. [16] T. Bal, M. von Krosigk, D.A. McCormick, Synaptic and membrane mechanisms underlying synchronized oscillations in the ferret lateral geniculate nucleus in vitro, J. of Physiology, vol. 483.3, pp. 641-663, 1995. [17] A. Destexhe, A. Babloyantz, T. Sejnowski, Ionic mechanisms for intrinsic slow oscillations in thalamic relay neurons, Biophysical Journal, vol.65, pp.1538-1552, 1993.
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A
Neuron inside
Is
In
-∑Ii -Is
Ii Vmem
Cmem
Neuron outside Ii
Is
: ith ionic current generator
: synaptic or stimulation current generator
B input pins
Structure of 1 ASIC
output pins I1
parameters I1 V mem1
ionic current generator
parameters I2
I2
ionic current generator
V mem2
Ica Calcium-dependence module
1: The Hodgkin-Huxley formalism. A: Equivalent electrical circuit of a neuron: the ionic currents Ii and the synaptic or stimulation current Is are summed on the membrane capacitance Cmem ; the voltage membrane Vmem is measured across Cmem . B: Typical structure of a model ASIC. The circuit contains 2 ionic current modules, and a Calcium dependence module (which can be inactivated by setting one dedicated parameter). To reproduce the Hodgkin-Huxley scheme, the I1, I2 , Vmem1 and Vmem2 pins are connected together, on an external Cmem capacitance, and on a gleak =Vleak circuitry. Voltages are applied on the parameter pins to set the model parameters, and the calcium-dependence circuit is inactivated. Fig.
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A
B
Conductance 1 Conductance 2 Calcium module
2: A: Microphotograph of a model ASIC; area including pads : 3mm2. B: The three rows of the core correspond respectively to 2 ionic current modules and 1 Calcium module. The input and output pads (parameters, Vmemi , Ii , ...) are on the border of the die. Fig.
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A
20 mV 1s
B
20 mV 1s
C
200 mV 1s
3: Membrane potential of a Pyloric Dilator (PD) neuron of the lobster stomatogastric Ganglion. A: Activity measurement on an in-vitro neuron. B: Numerical simulation of the PD model. C: Analog simulation of the same model using the circuits. Fig.
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A
t1
20 mV 0,2 s
B
200 mV 0,2 s
4: Membrane potential response of a thalamic nRt neuron following an hyperpolarizing stimulation at time t1. A: Numerical simulation of the nRt model. B: Analog simulation of the same model using the circuits and the same parameters. Fig.
19
A PD model
real
B
C model
real 1 sec
D
5: STG hybrid network using analog arti cial neurons. A: Network organization. B: Stable diphasic pattern. C: Increasing the gmaxCa parameter rises the Calcium current; a burst and tonic pattern appears. D: Rising gmaxCa gives a full bursting activity. Fig.
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stimulation (retina)
nRt
TC (cortex)
B TC t1
30 mV 1s
t2
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TC 30 mV 50 ms
t1
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6: Thalamus hybrid network using analog modeled neurons. A: Network organization; the external stimulation mimics optical information; the cortex would be a deeper layer connected to both nRt and TC cells. B: Network response to stimulation, with excitating-inhibiting sequence. C: Detail of one of those sequences, triggered by a stimulation at t1 and followed by spikes handshakes between the nRt and TC cells. Fig.
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P Cmem dVmem dt = , i Ii , IS , Ileak (1) Ileak = gleak (Vmem , Vleak ) (2) Ii = gioni (Vmem , Vequii ) (3) gioni = gmaxi mpihqi ; with p, q integers (4) mi dmdti = m1i , mi (5) m1i = 1+ Vmem,1 V offseti (6) Vslopei
d[dtCa] = K:ICa , [Ca] ; K constant (7) m1(Ca) = [Ca[Ca]+]C m1 Isyn = ,gsyn (Vmempost , Vequisyn ) (9) , with gsyn = f Vmempre Tab.
1: Hodgkin-Huxley formalism and Calcium-dependence equations.
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S. Le Masson, A. La aquière, Student Member IEEE, T. Bal and G. Le Masson S. Le Masson and A. La aquière, Laboratoire de Microélectronique IXL, CNRS UMR 5818, ENSERB, Université Bordeaux I, 351 cours de la Libération, 33405 Talence, France. T. Bal, Institut A. Fessard, CNRS UPR 2212, avenue de la Terrasse, 91198 Gif-surYvette, France. G. Le Masson, Institut de Neurosciences F. Magendie, unité INSERM 378, 1 rue C. Saint-Saëns 33077 Bordeaux, France.
Abstract Computational neuroscience is emerging as a new approach in biological neural networks studies. In an attempt to contribute to this eld, we present here a modeling work based on the implementation of biological neurons using speci c analog integrated circuits. We rst describe the mathematical basis of such models, then present analog emulations of dierent neurons. Each model is compared to its biological real counterpart as well as its numerical computation. Finally, we demonstrate the possible use of these analog models to interact dynamically with real cells through arti cial synapses within hybrid networks. This method is currently used to explore neural networks dynamics.
Key Words Analog circuits design, biological neurons, hybrid neural networks, neural modeling, stomatogastric ganglion, thalamus.
1
1 Introduction Biological neural networks are made of complex, highly non-linear elements, connected within networks containing thousands of interconnected neurons. The study of such networks, their properties and dynamic behavior could be limited by classical experimental approaches. One way to overcome these limitations is to study biological networks from a theoretical point-of-view, using mathematically de ned models of the neurons [1]. Various formalisms have been developped, and amongst them, the Hodgkin-Huxley's description of neuronal excitability [2], that has been shown to be one of the most fruitful and powerful framework for realistic neuronal modeling [3]. Computational neuroscience commonly uses it, by digital simulations on microcomputers or workstations. The well-known neurosimulator softwares [4] compute conductancebased models, which is a time-consuming task when the model complexity grows up. Consequently, real time computation is clearly di cult, even with the fastest microprocessors. More recently, were developed electronic analog models able to compute the HodgkinHuxley formalism [5], [6], [7]. They accurately reproduce the electrical properties of real neurons, as well as the time and voltage dependence of the membrane conductances. Moreover this electrical activity is replicated in real time. Computation speed could even be scaled-up by simply changing the time constants. However, all the analog models that have been developed present some level of abstraction, depending on the applications they are built for. We present in this paper the integrated circuits we designed speci cally for the realization of analog models that represent very detailed simulations of neural electrical activity. Indeed, our goal is to use these circuits as a tool for computational neuroscience. These speci c integrated circuits (ASICs) are designed in full custom mode using a bipolar and complementary MOS technology (BiCMOS). We will show how we reproduced the conductance-based structure of the mathematical model, and designed elementary blocks that can be programmed to emulate any conductance by an externally tuned set of parameters. A single chip contains several 2
conductance blocks, that can be independently connected to others to form a complex neuron or network. Finally, we will present the results of experimental work using `hybrid networks', i.e. networks interconnecting in real time modeled neurons and real ones in an biological preparation [8].
2 Realistic conductance-based models The electrical activity of a neuron is the consequence of the diusion of ionic species through its membrane [3]. This activity is characterized by a membrane potential, which is the voltage dierence between the outside and the inside of the cell. Ions ow through the cell membrane by ion-speci c channels, generating ionic currents. To each ion type is associated a reversal potential, due to the dierence between the intracellular and the extracellular concentrations, and following the Nernst equation. Ions tend to diuse through the membrane, attracting the membrane potential towards their respective reversal potential. For each ion type, the fraction of opened channels, which results from the interaction between time and voltage dependent activation and inactivation process, detrmine the global conductance of the membrane to that ion. Realistic conductance-based models of neurons are usually described using the Hodgkin-Huxley based approach [2]. In that formalism, the current owing across a membrane is integrated on the membrane capacitance, following the electrical equation (1), where Vmem is the membrane potential, Cmem the membrane capacitance, and IS an eventual external current (stimulation or synaptic input). A leak current Ileak is also considered, due to a small but permanent permeability of the membrane to some ions. The associated leak conductance gleak is generally non voltage dependent (2). Each ionic current Ii is described by a set of equations and parameters: Ii is the current passing through the ith channel type, given by equation (3) and (4), in which gmaxi is the maximal conductance value; mi and hi are the dynamic functions describing the permeability of membrane channels to this ion, represented by opening (activation mi) and closing (inactivation hi) fractions. As we see in equation (5), mi tends with a 3
time constant mi to its associated steady-state value m1i , which is a sigmoïdal function of Vmem . hi follows the same dynamics, with an inversely sigmoïdal steady-state value h1i . [Table 1 must be here] In ionic currents, additional expressions of interdependencies may in some case be necessary to express complex neural activities. Studies showed that some ionic currents do not only depend on the time and voltage variables, but also on intracellular calcium concentration. In that case, the corresponding activation (or inactivation) porcess is modi ed, as the calcium concentration changes. The calcium concentration is computed by a rst-order dierential equation depending on the ionic calcium current (equations (7) and (8)). All the parameters of the models following this formalism are experimentally determined, generally using voltage-clamp methods. They are then considered as biologically realistic models and usable in hybrid experiments (see section V). Synaptic transmission is another important concept of neural networks, as they are essential for information transmission between the cells. The synaptic current Isyn owing across the synapse is modeled as in equation (9), where Vmempre and Vmempost are respectively the presynaptic and postsynaptic membrane voltages. The synaptic conductance gsyn is described by the same formalism as votage-dependent channels (equations (4) to (6)). In hybrid experiments, one or more living neurons are connected to modeled neurons by arti cial synapses [8] and the real neuron may be either the presynaptic or postsynaptic. Thus we need to measure in real time its membrane voltage (in order to compute synaptic currents for both the living and modeled neurons) and to inject across the membrane a real synaptic current issued from the arti cial neuron. One solution is to use as an interface a specialized ampli er (Axoclamp 1B from Axon Instruments ) in a multiplexing mode (DCC for Discontinuous Current Clamp) allowing voltage measurement and current injection with a single glass intracellular 4
microelectrode impaled into the living neuron. That constraining structure (microelectrode + Axoclamp for each couple of synapses) makes experiments using those synaptic connections quite complex to handle, therefore only two or three couples of arti cial synapses can reasonably be considered in hybrid reconstructions.
3 Circuits implementation The circuits are ASICs (Application Speci c Integrated Circuits) designed with a 1.2 m BiCMOS technology, allowing the implementation on a chip of both bipolar and CMOS transistors. Bipolar transistors are used for the computation of sensitive functions (classically logarithmic, exponential or multiplicative). They also appear in blocks such as current conveyors or mirrors, to improve the precision and/or the output impedance. CMOS transistors help reducing the power consumption and minimize the chip area. Using that technology, we designed elementary modules [6], [7], which function is to behave as ionic current generators : they can be described as voltage-controlled current sources, with a transconductance replicating the Ii = f (Vmem ) function of the HodgkinHuxley formalism (with addition of the calcium-dependence equations). Due to technological constraints, we have a xed gain scaling Ii and Vmem variables. Real ionic currents in a neuron are nanoamperes, and membrane voltages are in the -100 mV to +100 mV range. Voltages on the chips have a gain of 10, and currents a gain of 1000, which implies a gain of 100 on conductance elements. We shall rst describe the implementation of a single module, i.e. an ionic current generator, that covers an area of approximately 1.5 mm x 2 mm. Mathematical functions such as addition, exponential, multiplication or derivation are computed by elementary circuits handling a small number of transistors. We minimized these numbers by designing the circuit in current mode [8]. All internal variables are treated as currents, and we can exploit the basic transfer functions of transistors that result in current outputs. However, this type of analog computation requires a very precise 5
design, as each single transistor in uences directly the transfer function; the circuits also has to be insensitive to the absolute sizing of transistors, which depends on the precision level of the technological implementation. The modules were then integrated on the ASICs in a 'full custom' mode. That mode of design allows an accurate sizing of each transistor, and ensures the relative sizing precision after fabrication. An analog model of a biological neuron is constructed as shown in gure 1, where each ionic current generator is as described above. A multi-conductances neuron is formed by externally connecting the current outputs on a membrane capacitance and a small circuit that computes the leak current (equation (2)). The measured membrane voltage is reinjected in the module as an equation variable. A single chip can actually include 2 or 3 conductances; each one has its own current output and Vmem input. A neuron is constructed by connecting a number of modules, coming from one or more chips. On the other side, modules on a single chip can be shared between dierent neurons, or be dedicated to synaptic connections, as we will see later. [Figure 1 must be here] One important speci city of our circuits is that none of the equations parameters is constraint. To each one is associated a chip input, on which the user applies a voltage that drives an internal voltage or current . These voltages or currents represent the Hodgkin-Huxley equation parameters, and correspond for each ionic current to the following terms: m1i (Vmem ); h1i (Vmem ); mi ; hi ; Vequii , gmaxi . To obtain such programmability, we separated each module is separated in several blocks as follows: 1. A sigmoïd generator is used to emulate the steady-state m1i and h1i functions (equation (7)); This function is de ned by an oset and a slope voltage. The Voffseti parameter is corresponding to the half-activation or half-inactivation potential, and Vslopei determines the slope of the linear part of the sigmoïd. The electronic representation of this function is based on a classical bipolar dierential pair, which current transfer function is a normalized sigmoïdal function. 6
The Voffset and Vslope parameters are introduced in the sigmoïdal function by a dierentiation and a multiplication applied on the input. 2. The output of the sigmoïd generator is connected to the next block, which computes equation (4) using a classical integrating scheme. The kinetic (identi ed with the m or h parameter) is set by a RC product, in which R is an on-chip resistor, and C an external capacitance, connected to a pin of the chip. We chose to leave the capacitance as an external component to be able to reproduce very slow kinetics, appearing in some models: they can go up to a few seconds, and in that case, the capacitance would be too large to be integrated. It should be noted that an approximation remains in the analog implementation where kinetics are not voltage-dependent, in contrast with the original model. We found out that voltage dependency of kinetics can be skipped in most models without degrading the result, if the kinetic value is carefully chosen. This is the case in the applications we present in sections IV and V. However, this non-dependence is a defect that cannot be neglected in all cases. Various implementation solutions are studied to solve that problem. 3. Finally the ionic current is the output of two cascaded multipliers, that generate successively gmaxi mpihqi and gioni (Vmem , Vequii ). These multipliers operate in current mode, and their output are normalized to respect the gain constraints we de ned at the beginning of the section (arti cial neuron currents = 100 times real neuron currents). The typical output range varies from -100 A to +100A. 4. The calcium-dependence equation is implemented in a separate module, designed to process equations (7) and (8), and to be interconnected with the ionic current generators. Its variable inputs are the ICa current, and an activation value m1 of the ionic current chosen as the calcium-dependent one. The output m1(Ca) is driven back to the source generator and input as the steady-state value m1 of equation (5). As before, and C are tunable parameters, using an external capacitance for the time constant. 7
[Figure 2 must be here]
4 Modeling neurons Computational neuroscientists have developed accurate models of neurons or neural networks presenting well-de ned behaviors. Model neural networks can faithfully reproduce the global activity resulting from the interaction of neurons within a real network, even though the simulated electrical activity of individual modeled neurons does not strictly t the electrical membrane activity of real neurons. This explains the slight dierences in electrical activity that may appear between real neurons and models. The results we present here only intend to prove the correct replication of the numerical model by an analog one. In fact, neuroscientists construct and validate models by determining the role of each parameter in shaping the individual cell response to synaptic or external stimulations, and by looking at the network overall behavior. To validate our own analog models, we directly compare the membrane potential coming from three sources: The experimental measurement on a living cell, through a microelectrode impaled into the neuron. The numerical computation of a mathematical model following the HodgkinHuxley formalism; this computation runs on a dedicated simulation software (`Maxim' software, by G. Le Masson; `Maxim' has been developed in C and runs on Apple Macintosh computers). The analog computation of the same model, using the ASICs. To further demonstrate the validity of our analog models, we present in the following paragraph some results of experiments where we con gured identical model chips with dierent parameters taken from diernet types of neurons. In each case, the membrane voltages measured from the analog model are compared with other voltage representations. The chips parameters are then the ones of the mathematical model, 8
only modi ed by the scale factors on currents and voltages indicated in section III. 4.1
Invertebrate neurons
The two-conductances model originally de ned by Hodgkin-Huxley runs spontaneous action potentials (spikes), using an inward Sodium current, with both activation and inactivation processes, and an outward Potassium current. However, most real neural cells present a more complex activity than simple spikes. In invertebrate species, a well-studied neural network is the lobster stomatogastric ganglion (STG) [9], [10]; one of the well identi ed cell of this network is called Pyloric Dilator (PD) and is modeled by a set of 7 ionic conductances, one of which is calcium-dependent. The resulting spontaneous activity is a succession of bursts of action potentials on top of a slowly oscillating envelop. This activity is quite sensitive to the precision of the equation computation and to the accuracy of the parameters, that have been experimentally de ned [11]: the equations have to be very well emulated by the ASICs to result in a correct membrane voltage. As previously mentionned, the only dierence with the original model is the absence of the voltage-dependence in the time constants. Figure (3 A) is a recording of a typical spontaneous bursting activity in a real PD cell. Comparing this reference activity to both the numerical model (3 B) and the integrated analog equivalent (3 C), reveals a good match of the overall shape and frequency of the signals. [Figure 3 must be here] 4.2
Vertebrate neurons
We also got interested in modeling neurons from vertebrates. We chose neurons from the thalamus for several reasons including the available accurate description of their ionic conductances, and the relative simplicity of the thalamic network. The thalamus is a grey matter nucleus that relays to the cortex most of the sensory information during awake state (including visual information), and becomes the generator of some types 9
of rythms (spindle waves) during early stages of sleep. Inside this structure, synaptic interactions between two populations of thalamocortical neurons are responsible for the generation of spindle waves that waxes and wanes periodically in the thalamocortical networks [12], [15]. The ability of that intra-thalamic network to produce a coordinated rhythmic activity could be a fundamental property involved in the gating of sensory information to the cortex and might be involved into paroxysmal synchronized oscillations resembling those underlying generalized absence seizures. We implemented a model of a neuron involved in such an intra-thalamic circuit, a nucleus Reticularis neuron, or nRt cell [13]. This model includes ve conductances, one of which is calcium-dependent. These conductances give the cell an endogeneous bursting behavior, occurring in response to intracellular negative current injection (4 A). Our analog model of a nRt neuron expresses the same behavior (4 B). [Figure 4 must be here]
5 Networks and hybrid networks The next step of the circuits evaluation process is to look at the emulation of neural networks, where neurons activity is driven by synaptic interactions. In that case, we use arti cial synapses implemented as indicated in section II, for the construction of hybrid networks [14]. In those systems, modeled neurons interact in real time with in vitro living cells; the hybrid system method gives the opportunity to access and tune separately the parameters of endogenous or synaptic conductances, and then to infer their individual role in shaping the entire network behavior. Modeled conductances or neurons are either added to the initial network, or replace a previously inactivated cell. Hybrid systems have been initially implemented using numerically-computed models, where the model computation speed is directly proportional to the number of conductances taken into account. The principle of the hybrid systems implies that the model runs in real time; one understands easily that limitations, due to the numerical computation time, quickly appear when the model complexity grows up. Analog com10
putation de nitely solves the problem of real time, provided that time constants tuned on the chips are identical to those of the real neurons. As for the isolated modeled cells, we worked with hybrid systems on vertebrate and invertebrate neural networks. Hybrid networks were constructed using the models described in section IV (PD and nRt neurons), as these models have been perfectly characterized and validated [13], [14]. 5.1
Invertebrate neural network
We will consider at rst an hybrid system, which wiring diagram is shown in 5 A. A PD model is connected to a spontaneously bursting neuron, recorded from an invertebrate stomatogastric ganglion. All synapses are inhibitory. Figure 5 B to 5 D show the eect of a linear increase of the calcium current in the model (by the mean of an increase of the gmaxCa parameter). Without calcium current, the model behaves as a tonic cell and res action potentials. When connected with reciprocally inhibiting synapses, the two cells organize a diphasic rhythm of alternate bursts (5 B). If the calcium maximal conductance is increased, the model expresses a more complex response to the real neuron's synaptic inhibition: for an intermediate value of gmaxCa , the model neuron responds by a single burst followed by a tonic phase (5 C), whereas for stronger values, it has a full bursting activity (5 D). In this latter case, the network is stable with a 3 to 1 entrainment. In all cases, one can note the synaptic hyperpolarization in the real neuron, due to the in uence of bursts and action potentials in the model neuron. [Figure 5 must be here] 5.2
Vertebrate neural network
Our method is also particularly well suited for constructing hybrid networks using vertebrate brain slices. In such preparations, due to slice thickness (400 m), long range synaptic connections are often cut, except for very local ones. In the visual thalamus, 11
relay cells (also named TC for thalamocortical) directly receive inputs from the retina. Their connections in a feed forward way to the visual cortex allow the visual inputs to be transmitted to the dedicated treatment areas. But within the thalamus, the TC cells are also connected to another thalamic nucleus (nucleus Reticularis), and its inhibitory nRt cells [15]. The global synaptic diagram between TC and nRt cells is a simple excitatory (from TC to nRt) and inhibitory (from nRt to TC) loop, as shown in 6 A. Depending on the state of this neuronal loop, visual ow can be accurately transmitted to the cortex (awake state), or actively ltered by strong oscillations (sleep-like state). These oscillations are the result of the synaptic interactions between the neurons, but also of their intrinsic cellular properties. In only one specie, the ferret, synaptic connections between TC and nRt cells are well preserved after slicing. Working on other species (rats or guinea pigs), and using the hybrid networks method with an analog model of a nRt cell, we were able to reconstruct a functional circuit that generates sleep-like oscillations (spindle waves). (6 B) and (6 C) show the resulting activity of our arti cial network, in response to a short negative current injection in the TC neuron (time t1 ). This stimulation triggers a long lasting (0.5 s to 1 s) oscillation of the membrane potential in both cells. The oscillation sequence is based on the following cellular and synaptic mechanisms: the negative current step injection induces a rebound calcium potential and a burst of spikes in the TC cell resulting in the synaptic excitation of the modeled nRt cell. In turn, the ring of the nRt cell results in strong inhibitions of the TC cell leading to more rebound calcium potentials. Due to that handshake sequence, the system nally oscillates at a 12 to 14 Hz frequency, until a slow calcium-dependent current terminates the wave (time t2). This simple experiment illustrate how hybrid systems can facilitate the analysis of network activity in experimental situations where classical approaches are not possible. [Figure 6 must be here]
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6 Conclusion We have presented in this paper a summary of the design principles and applications of a set of analog integrated circuits that we developed to emulate mathematical models of biological neurons. That type of circuits is an interesting evolution in computational neuroscience, due to the following characteristics: - on-chip programmability of the model parameters, in order to use identical circuit in dierent con gurations to emulate dierent neurons. - better circuit integration (i.e. smaller die size) than an equivalent digital implementation. - intrinsic real time behavior, as circuits are an electronic equivalent of the mathematical description (including the time constants). That latest property is exploited in the hybrid networks experiments, where neural networks are reconstructed using both arti cial and living neurons. For a purely computational application, including for example regulation functions (that are very important for the evaluation of functional stability in neural networks), it is very easy to accelerate the computation speed of the electronic models, only by multiplying all the time constants by an identical factor. Cut-o frequencies of the used BiCMOS technology for our type of design is high enough (around 50 MHz) compared to the original model one (a few kHz) to allow a computing scaling-up by a factor of at least 1000. In comparison, the numerical computation of the electrical activity of a nRt cell modeled as in section IV B could not be done with a sampling rate better than 10 kHz, using an optimized version of the neurosimulator software 'Maxim' running on a 132 MHz Apple computer (Power Macintosh 9500).
REFERENCES [1] B. Softkey, C. Koch, Single cell models, in M. Arbib, editor, The handbook of brain theory and neural networks, pp. 879-884, MIT Press, Boston, MA, 1995. [2] A.L. Hodgkin and A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, Journal of Physiology, vol. 117, pp. 500-544, 1952. [3] C. Koch and I. Segev, Editors, Methods 13 in neuronal modeling: from synapses to
networks, MIT Press, Cambridge, MA, 1989. [4] J. Murre, Neurosimulators, in M. Arbib, editor, The handbook of brain theory and neural networks, pp. 634-639, MIT Press, Boston, MA, 1995. [5] C. A. Mead, Analog VLSI and neural systems, Addison Wesley, Reading, MA, 1989. [6] R. Douglas and M. Mahowald, A construction set for silicon neurons, in Neural and Electronics Networks, F. Zornetzer, Editor, Academic Press, Arlington, 1995. [7] E. A. Vittoz and X. Arreguit, Systems based on bio-inspired analog VLSI, MicroNeuro'96 tutorial, Lausanne, Feb. 12-14, 1996. [8] G. Le Masson, S. Le Masson and M. Moulins, From conductances to neural networks properties: analysis of simple circuits using the hybrid networks method, Progress in Biophysics and Molecular Biology, vol.64 n 2/3, pp. 201-220, 1995. [9] D. Dupeyron, S. Le Masson, Y. Deval, G. Le Masson and J.P. Dom, A BiCMOS implementation of the Hodgkin-Huxley formalism, Proc. of MicroNeuro'96, Lausanne, IEEE Computer Society Press, pp. 311-316, 1996. [10] A. La aquière, S. Le Masson, D. Dupeyron and G. Le Masson, Analog circuits emulating biological neurons in real-time experiments, IEEE International Conference on Biomedical Engineering (EMBS'97, Chicago, IL), pp. 2035-2038, October 1997. [11] C. Toumazou, F.G. Lidgey and D.G. Haigh, Analog IC design: a current-mode approach, IEE circuits and systems, London, 1990. [12] J. Golowasch, F. Buchholtz, IR. Epstein, E. Marder, Contribution of individual ionic currents to activity of a model stomatogastric ganglion neuron, Journal of Neurophysiology, vol. 67, pp. 341-349, 1992. [13] R.M. Harris-Warrick, E. Marder, A.I. Selverston and M. Moulins, editors, Dynamic biological networks: the stomatogastric nervous system, Cambridge, MA, MIT Press, 1992. [14] G. Le Masson, E. Marder and L.F. Abbott, Activity-dependent regulation of conductances in model neurons, Science, vol. 259, pp. 1915-1917, 1993. [15] M. von Krosigk, T. Bal, D.A. McCormick, Cellular mechanisms of a synchro14
nized oscillation in the thalamus, Science, vol. 261, pp. 361-364, 1993. [16] T. Bal, M. von Krosigk, D.A. McCormick, Synaptic and membrane mechanisms underlying synchronized oscillations in the ferret lateral geniculate nucleus in vitro, J. of Physiology, vol. 483.3, pp. 641-663, 1995. [17] A. Destexhe, A. Babloyantz, T. Sejnowski, Ionic mechanisms for intrinsic slow oscillations in thalamic relay neurons, Biophysical Journal, vol.65, pp.1538-1552, 1993.
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A
Neuron inside
Is
In
-∑Ii -Is
Ii Vmem
Cmem
Neuron outside Ii
Is
: ith ionic current generator
: synaptic or stimulation current generator
B input pins
Structure of 1 ASIC
output pins I1
parameters I1 V mem1
ionic current generator
parameters I2
I2
ionic current generator
V mem2
Ica Calcium-dependence module
1: The Hodgkin-Huxley formalism. A: Equivalent electrical circuit of a neuron: the ionic currents Ii and the synaptic or stimulation current Is are summed on the membrane capacitance Cmem ; the voltage membrane Vmem is measured across Cmem . B: Typical structure of a model ASIC. The circuit contains 2 ionic current modules, and a Calcium dependence module (which can be inactivated by setting one dedicated parameter). To reproduce the Hodgkin-Huxley scheme, the I1, I2 , Vmem1 and Vmem2 pins are connected together, on an external Cmem capacitance, and on a gleak =Vleak circuitry. Voltages are applied on the parameter pins to set the model parameters, and the calcium-dependence circuit is inactivated. Fig.
16
A
B
Conductance 1 Conductance 2 Calcium module
2: A: Microphotograph of a model ASIC; area including pads : 3mm2. B: The three rows of the core correspond respectively to 2 ionic current modules and 1 Calcium module. The input and output pads (parameters, Vmemi , Ii , ...) are on the border of the die. Fig.
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A
20 mV 1s
B
20 mV 1s
C
200 mV 1s
3: Membrane potential of a Pyloric Dilator (PD) neuron of the lobster stomatogastric Ganglion. A: Activity measurement on an in-vitro neuron. B: Numerical simulation of the PD model. C: Analog simulation of the same model using the circuits. Fig.
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t1
20 mV 0,2 s
B
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4: Membrane potential response of a thalamic nRt neuron following an hyperpolarizing stimulation at time t1. A: Numerical simulation of the nRt model. B: Analog simulation of the same model using the circuits and the same parameters. Fig.
19
A PD model
real
B
C model
real 1 sec
D
5: STG hybrid network using analog arti cial neurons. A: Network organization. B: Stable diphasic pattern. C: Increasing the gmaxCa parameter rises the Calcium current; a burst and tonic pattern appears. D: Rising gmaxCa gives a full bursting activity. Fig.
20
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stimulation (retina)
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30 mV 1s
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6: Thalamus hybrid network using analog modeled neurons. A: Network organization; the external stimulation mimics optical information; the cortex would be a deeper layer connected to both nRt and TC cells. B: Network response to stimulation, with excitating-inhibiting sequence. C: Detail of one of those sequences, triggered by a stimulation at t1 and followed by spikes handshakes between the nRt and TC cells. Fig.
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P Cmem dVmem dt = , i Ii , IS , Ileak (1) Ileak = gleak (Vmem , Vleak ) (2) Ii = gioni (Vmem , Vequii ) (3) gioni = gmaxi mpihqi ; with p, q integers (4) mi dmdti = m1i , mi (5) m1i = 1+ Vmem,1 V offseti (6) Vslopei
d[dtCa] = K:ICa , [Ca] ; K constant (7) m1(Ca) = [Ca[Ca]+]C m1 Isyn = ,gsyn (Vmempost , Vequisyn ) (9) , with gsyn = f Vmempre Tab.
1: Hodgkin-Huxley formalism and Calcium-dependence equations.
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