Elliot D. Menschik, Shih-Cheng Yen, and Leif H. Finkel

Transcription

1 To appear in Computational Neuroscience: Trends in Research, 1998 (J.M. Bower, ed.) ATTRACTOR DYNAMICS IN REALISTIC HIPPOCAMPAL NETWORKS Elliot D. Menschik, Shih-Cheng Yen, and Leif H. Finkel Institute of Neurological Sciences and Department of Bioengineering University of Pennsylvania 332 Smith Walk Philadelphia, PA 1914 INTRODUCTION Autoassociative attractor neural networks 1,2 provide a powerful paradigm for the storage and recall of memories, however, their biological plausibility has always remained in question. Given the complexity and variability of biological networks, it seems likely that a biological attractor network must be regulated by some control structure. We describe a functional architecture for implementing the Hopfield attractor paradigm in a model of the CA3 region of the hippocampus. In this model the control structure is provided by the intrinsic and extrinsic rhythms (gamma and theta) of the hippocampus and neuromodulatory input. Cellular-level simulations are shown that serve as a proof-of-concept. The function of the model is demonstrated in the left-hand panel of Figure 1. Shown are hypothetical spike traces of CA3 pyramidal cells with idealized gamma and theta population rhythms shown for reference. The state of the network is defined by the spatial pattern of temporally-precise spikes during the time window defined by a single gamma cycle. A memory is reached when the network converges to a fixed-point attractor. The gamma-band synchronization is induced by a network of mutually inhibitory interneurons 3-5. Theta-band oscillations, induced by septal interneurons, act to clock new perforant input to the network from entorhinal cortex, terminate each attractor state, and reset the network for the next set of entorhinal inputs. Lastly, cholinergic input from the medial septum is responsible for maintaining otherwise bursting pyramidal cells in a single spike firing mode. As such, this model provides some putative roles for hippocampal interneurons, synchronous oscillations, and cholinergic neuromodulation. The CA3 region provides an ideal neural substrate for an autoassociative network 6. In fact, numerous investigators have used the autoassociative model to create functional hippocampal networks for memory and spatial navigation, but, to date, nearly all have relied upon simplified models of individual neurons. Such studies have provided both insight and novel theories of hippocampal function. However, more detailed and realistic, compartmental models are necessary for studying the effects of normal and pathological cellular mechanisms on network function. Our study is directed to the following questions: Can networks of biological neurons perform the same or analogous computations as networks of artificial (i.e. formal, mathematical) neurons? What are the functional consequences of neuromodulation upon network dynamics? How might pathological perturbations at the cellular and subcellular levels translate into network dysfunction at a functional level? METHODS Compartmental simulations were constructed using PGENESIS 7, the recent parallel implementation of the GENESIS development package. Simulations were performed on a network of 18 Silicon Graphics Indy workstations. Differential equations were solved using Crank-Nicholson implicit integration with a step size of 25µs. 1

2 attractor state Entorhinal cortex pyramidal cell 1 pyramidal cell 2 pyramidal cell 3 pyramidal cell 4 pyramidal cell 5 pyramidal cell 6 pyramidal cell 7 Medial Septal Nucleus Cholinergic cells CA3 Pyramidal cell network Pyramidal cells pyramidal cell N Theta-burst cells Cell Morphologies gamma rhythm theta rhythm s. l.-m. s. r. s. l. s. p. s. o. perforant input time Neurotransmitters Glutamate GABA Acetylcholine Basket cell network Chandelier cells pyramidal interneuron Figure 1. (Left) Operation of the biological analog of an autoassociative attractor neural network. (Right) The architecture and behavior of the model is consistent with known hippocampal anatomy and physiology. Shown is a network schematic and sample simulated somatic voltage traces from the three CA3 cell populations. s. l.-m., stratum lacunosum-moleculare; s. r., stratum radiatum; s. l., stratum lucidum; s. p., stratum pyramidale; s. o., stratum oriens. Pyramidal cells, interneurons, and synapses The cellular models chosen for the simulations are the most highly detailed and realistic hippocampal cells developed to date: the 66-compartment hippocampal CA3 pyramidal and the 51-compartment hippocampal interneuron developed by Traub and colleagues 8,9. For the most part, AMPA, NMDA, and GABA A synapses were implemented as described by Traub and colleagues 1. Network connectivity This preliminary network consists of 24 neurons (8 pyramidal cells, 8 basket cells, and 8 chandelier cells). Its design is inspired by the known anatomy of CA3 and is sketched in the right-hand panel of Figure 1. Mutually inhibitory basket cells are depolarized by septal cholinergic and local pyramidal cell glutamatergic input while being simultaneously inhibited by septal perisomatic GABAergic input oscillating at theta frequencies. The inhibition between basket cells creates synchronous gamma oscillations 3-5 which are themselves modulated at theta frequencies by the septal inhibition. CA3 pyramidal cells are depolarized in the s. p. by septal cholinergic input, in the s. r. by recurrent glutamatergic input, and in the s. l.-m. by perforant glutamatergic input from entorhinal cortex. Synchronous oscillatory inhibition from the basket cells in the perisomatic region constrains pyramidal cell firing while recurrent inhibition from chandelier cells at the axonal initial segment balances recurrent excitation. Two random 8-bit patterns (1111 and 11111) were chosen and stored in the recurrent synaptic matrix using Hopfield s original algorithm 1 to scale maximal synaptic conductances. Cholinergic neuromodulation Neuromodulation via acetylcholine (ACh) is implemented at the cellular level inhibiting intrinsic membrane currents and diffusely depolarizing cells. For ionic current inhibition we derived dose-response curves based on a Michaelis-Menten model. Nonlinear curve fits matched very closely the data of Madison et al. 11 for I AHP and Toselli and Lux 12 for I Ca. Diffuse cholinergic depolarization of pyramidal and basket cells is modeled indirectly using depolarizing somatic current injection. Additional detail is provided in a recent, larger-scale study 13. 2

3 RESULTS Network function Before examining the behavior of the compartmental network model, our first step was to establish control conditions by exploring the trajectories of all 256 initial states of an 8-cell artificial Hopfield attractor network storing the randomly chosen binary patterns 1111 and (9C and AB in hexadecimal notation). For each initial state we recorded the subsequent network states at each time step using synchronous updating. A map of these trajectories is shown in Figure 2. With the control established, each of the 256 initial states was presented to the biologically-based network as simulated entorhinal input on the perforant pathway (i.e. transient glutamatergic excitation of the distal apical arbor). After the initial spike pattern, the network state was allowed to evolve according to its recurrent connectivity. A sample spike trace for the pyramidal cell network is shown in Figure 3 with the inputs 11 (18 hex) and 111 (19 hex). The attractors reached are in fact the same stored patterns as found in the artificial network. While it is not clear from this data that the final state reached is an attractor, additional simulations show that if the theta rhythm is interrupted (i.e. the pyramidal cells are not inhibited) at the end of a theta cycle, the firing pattern is indeed stable (data not shown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igure 2. Trajectories of the 256 initial states of the artificial attractor network storing the binary patterns and 1111 (AB and 9C, respectively, in hexadecimal notation). By symmetry the inverse patterns are also attractors. All initial states move to one of the four attractors within 2 time steps. 3

4 18 9C 19 AB cell 1 cell 2 cell 3 cell 4 cell 5 cell 6 cell 7 cell 8 gamma theta Figure 3. Performance of the biological attractor network. Shown are spike traces from the 8 pyramidal cells following two entorhinal inputs at the onset of each theta cycle. Idealized gamma and theta population oscillations are shown for reference. A comparison with Figure 2 shows that the correct attractors are reached by the end of the theta cycle, but that different trajectories are taken than in the artificial attractor network. The hexadecimal equivalents of the network inputs and outputs are shown at the top. 28 ms Cholinergic neuromodulation of single pyramidal cells Acetylcholine plays a critical role in the biological network model not only by providing diffuse excitation of the cells, but by regulating the firing mode of pyramidal cells. Shown in the top panel of Figure 4 is the effect of varying [ACh] for the pyramidal cell model using the dose-response curves discussed in the Methods section. Somatically-recorded voltage traces are shown for the same cell using simulated application of.1 to 1 µm ACh. As [ACh] rises, the cell undergoes a marked transition from lowfrequency bursting to high-frequency spiking. In contrast to the pyramidal cell model, the effects of ACh on the fast spiking of the interneuron model is negligible (data not shown). The transition of the intrinsically bursting pyramidal cell to a spiking regime can be qualitatively understood by recognizing that I Ca is responsible for the slow depolarization underlying the burst and also the interplay with the fast sodium current that causes the rapid series of action potentials riding the slow depolarization. In contrast, I AHP is the current responsible for terminating the burst and maintaining a long hyperpolarization following it. The inhibition of I Ca by ACh removes the slow calcium-dependent wave, diminishes the reverberating depolarization between soma and adjacent dendrites that create the rapid series of overlying spikes, and affects the calcium dependency of I AHP. The inhibition of I AHP by ACh markedly reduces the afterhyperpolarization that terminates the burst and maintains the low inter-burst interval. It should also be noted that under these conditions, a very small depolarizing current (.1 na) results in a very high spike rate for pyramidal cells as the simulated concentration of ACh rises. This rise in spike frequency can be (and in our simulations is) held in check by interneuronal control. Finally, while our simulations demonstrate that cholinergic input is sufficient to induce a transition in pyramidal cell firing mode, Traub and colleagues have shown that tonic somatic current injections 8,14 or tonic stimulation of slow dendritic GABA A receptors 8 may be responsible for a similar functional shift. Our results are independent of these two factors as we have held current injection constant at a level that does not induce spiking in the absence of ACh, and we have not included slow dendritic GABA A receptors. Our simulations also suggest that bursting and spiking behavior in hippocampal pyramidal cells have distinct advantages that may be exploited by switching between these two firing modes. The bottom panels of Figure 4 plot the calcium concentration in the pyramidal cell model as a function of time and space for a spiking cell and a bursting cell. The data shows that backpropagating single spikes are quite poor at inducing calcium influx in passive distal dendrites in contrast to backpropagating bursts. 4

5 .5 um.1 um um um um um.5 1 um.5.5 seconds [Ca] (mm) 1 5 [Ca] (mm) time (ms) layer Figure 4. Muscarinic neuromodulation of a pyramidal cell. (Top) Shown are 2.5 seconds of simulated somatic recordings from the model pyramidal cell at various levels of cholinergic input. The cell received.1 na current injection in all traces. (Bottom) Compartmental distribution of calcium concentration in mm as a function of time in spiking (left) and bursting (right) hippocampal pyramidal cells. Layers 1-3 are the basal dendrites, layer 4 is the soma, and layers 5-11 are the apical dendritic arbor. 8 6 time (ms) layer 8 1 DISCUSSION The biological attractor network It should be noted that the results presented here are of a preliminary nature and our primary goal was to present a concept for biological attractor networks, describe the structure of such a network, and demonstrate its practical feasibility. Nevertheless, use of only eight pyramidal cells, while facilitating the comparison with an artificial attractor network as well as making practical the simulation on a local network of workstations, is a serious limitation. Storing just two patterns heavily overloads the storage capacity of such a small network. A much larger version of this network has since demonstrated the scalability of the model 13. Another problem encountered by the network is the relatively frequent (approximately 4%) failure to converge to the appropriate attractor in the time window provided by the theta rhythm. This flaw is due to the small size of the network and the relatively slow gamma-band rhythm produced by the basket cells. Again, a larger network with more gamma cycles per theta cycle has shown nearly perfect convergence to any of several stored patterns even in the presence of significant noise 13. Acetylcholine and the functional role of spikes and bursts Physiological recordings show that hippocampal pyramidal cells are capable of either bursting or spiking, and that these different modes are correlated with the behavioral state of the rat 15. From a theoretical perspective, these two firing modes have their respective advantages. Spiking is rapid and can have a temporal precision of a millisecond or so, allowing for efficient representation of information and a 5

6 well-defined network activity state. However, our simulations have demonstrated that spiking may be poorly suited for inducing and/or maintaining synaptic plasticity in the distal dendritic arbor. Rather, this function appears to be better fulfilled by backpropagating bursts which are capable of causing significant alterations of calcium levels in the dendrites. The drawback for hippocampal bursts lies in their typical low-frequency and variable length which make the representation of information difficult and inefficient at best. Together with the behavioral correlations, our findings suggest that spiking behavior is necessary for the initial processing of novel information and its later recall, while bursting is necessary for more permanent storage of patterns via the induction of LTP and LTD. This view is wholly consistent with Buzsáki s twostage memory model 16. Our model indicates that ACh acts on at least two levels to initiate and manage a transition from bursting to spiking behavior, at least in intrinsically bursting hippocampal pyramidal cells. At the cellular level, the transition is due to a reduction in the afterhyperpolarizing calcium-dependent potassium current and the high-threshold calcium current. At the network level, the diffuse depolarizing action of ACh on interneurons can serve as the driving force for mutually inhibitory interneuronal networks which can generate gamma-band rhythmicity and thereby control the timing of pyramidal cell spiking. ACKNOWLEDGEMENTS We thank Nigel Goddard and Greg Hood at the Pittsburgh Supercomputing Center for providing a betatest version of PGENESIS and help with its implementation. Supported by grants from Mrs. Patricia Kind, The Whitaker Foundation, and the Office of Naval Research. REFERENCES 1. J.J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proceedings of the National Academy of Sciences of the United States of America, 79: (1982). 2. D.J. Amit, Modeling Brain Function: The world of attractor neural networks, Cambridge University Press, New York, X.J. Wang and G. Buzsáki, Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model, Journal of Neuroscience, 16: (1996). 4. R.D. Traub, M.A. Whittington, I.M. Stanford and J.G.R. Jefferys, A mechanism for generation of long-range synchronous fast oscillations in the cortex, Nature, 383:621-4 (1996). 5. P. Bush and T.J. Sejnowski, Inhibition synchronizes sparsely connected cortical neurons within and between columns in realistic network models, Journal of Computational Neuroscience, 3:91-11 (1996). 6. A. Treves and E.T. Rolls, Computational analysis of the role of the hippocampus in memory, Hippocampus, 4: (1994). 7. N.H. Goddard and G. Hood, Large Scale Simulation with PGENESIS. In J. M. Bower and D. Beeman (Eds.), The Book of GENESIS: Exploring Realistic Neural Models with the GEneral NEural SImulation System, Springer-Verlag, in press. 8. R.D. Traub, J.G.R. Jefferys, R. Miles, M.A. Whittington and K. Tóth, A branching dendritic model of a rodent CA3 pyramidal neurone, Journal of Physiology, 481:79-95 (1994). 9. R.D. Traub and R. Miles, Pyramidal cell-to-inhibitory cell spike transduction explicable by active dendritic conductances in inhibitory cell, Journal of Computational Neuroscience, 2:291-8 (1995). 1. R.D. Traub, M.A. Whittington, S.B. Colling, G. Buzsáki and J.G.R. Jefferys, Analysis of gamma rhythms in the rat hippocampus in vitro and in vivo, Journal of Physiology, 493: (1996). 11. D.V. Madison, B. Lancaster and R.A. Nicoll, Voltage clamp analysis of cholinergic action in the hippocampus, Journal of Neuroscience, 7: (1987). 12. M. Toselli and H.D. Lux, GTP-binding proteins mediate acetylcholine inhibition of voltage dependent calcium channels in hippocampal neurons, Pflugers Archiv - European Journal of Physiology, 413: (1989). 13. E.D. Menschik and L.H. Finkel, Neuromodulatory control of hippocampal function: Towards a model of Alzheimer's disease, Artificial Intelligence in Medicine, Special Issue: Computational Modeling of Brain Disorders (in press). 14. R.D. Traub, R.K. Wong, R. Miles and H. Michelson, A model of a CA3 hippocampal pyramidal neuron incorporating voltageclamp data on intrinsic conductances, Journal of Neurophysiology, 66:635-5 (1991). 15. R.D. Traub and R. Miles, Neuronal Networks of the Hippocampus, Cambridge University Press, New York, G. Buzsáki, Two-stage model of memory trace formation: a role for "noisy" brain states, Neuroscience, 31:551-7 (1989). 6