To appear in Computational Neuroscience: Trends in Research, 1998 (J.M. Bower, ed.)

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ATTRACTOR DYNAMICS IN REALISTIC HIPPOCAMPAL NETWORKS

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

Institute of Neurological Sciences and Department of BioengineeringUniversity of Pennsylvania3320 Smith WalkPhiladelphia, PA 19104

INTRODUCTION

Autoassociative attractor neural networks1,2 provide a powerful paradigm for the storage and recall ofmemories, however, their biological plausibility has always remained in question. Given the complexity andvariability of biological networks, it seems likely that a “biological” attractor network must be regulated bysome control structure. We describe a functional architecture for implementing the Hopfield attractorparadigm in a model of the CA3 region of the hippocampus. In this model the control structure is providedby 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 hypotheticalspike 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 windowdefined by a single gamma cycle. A memory is reached when the network converges to a fixed-pointattractor. The gamma-band synchronization is induced by a network of mutually inhibitory interneurons3-5.Theta-band oscillations, induced by septal interneurons, act to clock new perforant input to the network fromentorhinal 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 pyramidalcells in a single spike firing mode. As such, this model provides some putative roles for hippocampalinterneurons, synchronous oscillations, and cholinergic neuromodulation.

The CA3 region provides an ideal neural substrate for an autoassociative network6. In fact, numerousinvestigators have used the autoassociative model to create functional hippocampal networks for memory andspatial navigation, but, to date, nearly all have relied upon simplified models of individual neurons. Suchstudies have provided both insight and novel theories of hippocampal function. However, more detailed andrealistic, compartmental models are necessary for studying the effects of normal and pathological cellularmechanisms on network function. Our study is directed to the following questions: Can networks ofbiological 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 dysfunctionat a functional level?

METHODS

Compartmental simulations were constructed using PGENESIS7, the recent parallel implementation ofthe GENESIS development package. Simulations were performed on a network of 18 Silicon Graphics Indyworkstations. Differential equations were solved using Crank-Nicholson implicit integration with a step sizeof 25µs.

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Pyramidal cells, interneurons, and synapses

The cellular models chosen for the simulations are the most highly detailed and realistic hippocampalcells developed to date: the 66-compartment hippocampal CA3 pyramidal and the 51-compartmenthippocampal interneuron developed by Traub and colleagues8,9. For the most part, AMPA, NMDA, andGABAA synapses were implemented as described by Traub and colleagues10.

Network connectivity

This preliminary network consists of 24 neurons (8 pyramidal cells, 8 basket cells, and 8 chandeliercells). Its design is inspired by the known anatomy of CA3 and is sketched in the right-hand panel of Figure1. Mutually inhibitory basket cells are depolarized by septal cholinergic and local pyramidal cellglutamatergic input while being simultaneously inhibited by septal perisomatic GABAergic input oscillatingat theta frequencies. The inhibition between basket cells creates synchronous gamma oscillations3-5 whichare themselves modulated at theta frequencies by the septal inhibition. CA3 pyramidal cells are depolarizedin the s. p. by septal cholinergic input, in the s. r. by recurrent glutamatergic input, and in the s. l.-m. byperforant glutamatergic input from entorhinal cortex. Synchronous oscillatory inhibition from the basketcells in the perisomatic region constrains pyramidal cell firing while recurrent inhibition from chandeliercells at the axonal initial segment balances recurrent excitation. Two random 8-bit patterns (10011100 and10101011) were chosen and stored in the recurrent synaptic matrix using Hopfield’s original algorithm1 toscale maximal synaptic conductances.

Cholinergic neuromodulation

Neuromodulation via acetylcholine (ACh) is implemented at the cellular level inhibiting intrinsicmembrane currents and diffusely depolarizing cells. For ionic current inhibition we derived dose-responsecurves based on a Michaelis-Menten model. Nonlinear curve fits matched very closely the data ofMadison et al.11 for IAHP and Toselli and Lux12 for ICa. Diffuse cholinergic depolarization of pyramidal andbasket cells is modeled indirectly using depolarizing somatic current injection. Additional detail is providedin a recent, larger-scale study13.

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Figure 1 . (Left) Operation of the biological analog of an autoassociative attractor neural network. (Right) The architecture andbehavior of the model is consistent with known hippocampal anatomy and physiology. Shown is a network schematic and samplesimulated somatic voltage traces from the three CA3 cell populations. s. l.-m., stratum lacunosum-moleculare; s. r., stratumradiatum; s. l., stratum lucidum; s. p., stratum pyramidale; s. o., stratum oriens.

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RESULTS

Network function

Before examining the behavior of the compartmental network model, our first step was to establishcontrol conditions by exploring the trajectories of all 256 initial states of an 8-cell artificial Hopfield attractornetwork storing the randomly chosen binary patterns 10011100 and 10101011 (9C and AB in hexadecimalnotation). For each initial state we recorded the subsequent network states at each time step usingsynchronous 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-basednetwork as simulated entorhinal input on the perforant pathway (i.e. transient glutamatergic excitation of thedistal apical arbor). After the initial spike pattern, the network state was allowed to evolve according to itsrecurrent connectivity. A sample spike trace for the pyramidal cell network is shown in Figure 3 with theinputs 00011000 (18 hex) and 00011001 (19 hex). The attractors reached are in fact the same storedpatterns as found in the artificial network. While it is not clear from this data that the final state reached is anattractor, additional simulations show that if the theta rhythm is interrupted (i.e. the pyramidal cells are notinhibited) at the end of a theta cycle, the firing pattern is indeed stable (data not shown).

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Figure 2. Trajectories of the 256 initial states of the artificial attractor network storing the binary patterns 10101011 and10011100 (AB and 9C, respectively, in hexadecimal notation). By symmetry the inverse patterns are also attractors. All initialstates move to one of the four attractors within 2 time steps.

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Cholinergic neuromodulation of single pyramidal cells

Acetylcholine plays a critical role in the biological network model not only by providing diffuseexcitation of the cells, but by regulating the firing mode of pyramidal cells. Shown in the top panel ofFigure 4 is the effect of varying [ACh] for the pyramidal cell model using the dose-response curves discussedin the Methods section. Somatically-recorded voltage traces are shown for the same cell using simulatedapplication of 0.1 to 100 µM ACh. As [ACh] rises, the cell undergoes a marked transition from low-frequency bursting to high-frequency spiking. In contrast to the pyramidal cell model, the effects of ACh onthe 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 qualitativelyunderstood by recognizing that ICa is responsible for the slow depolarization underlying the burst and alsothe interplay with the fast sodium current that causes the rapid series of action potentials riding the slowdepolarization. In contrast, IAHP is the current responsible for terminating the burst and maintaining a longhyperpolarization following it. The inhibition of ICa by ACh removes the slow calcium-dependent wave,diminishes the reverberating depolarization between soma and adjacent dendrites that create the rapid seriesof overlying spikes, and affects the calcium dependency of IAHP. The inhibition of IAHP by ACh markedlyreduces 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 (0.1 nA) results ina very high spike rate for pyramidal cells as the simulated concentration of ACh rises. This rise in spikefrequency can be (and in our simulations is) held in check by interneuronal control. Finally, while oursimulations demonstrate that cholinergic input is sufficient to induce a transition in pyramidal cell firingmode, Traub and colleagues have shown that tonic somatic current injections8,14 or tonic stimulation of slowdendritic GABAA receptors8 may be responsible for a similar functional shift. Our results are independent ofthese two factors as we have held current injection constant at a level that does not induce spiking in theabsence of ACh, and we have not included slow dendritic GABAA receptors.

Our simulations also suggest that bursting and spiking behavior in hippocampal pyramidal cells havedistinct advantages that may be exploited by switching between these two firing modes. The bottom panelsof Figure 4 plot the calcium concentration in the pyramidal cell model as a function of time and space for aspiking cell and a bursting cell. The data shows that backpropagating single spikes are quite poor atinducing calcium influx in passive distal dendrites in contrast to backpropagating bursts.

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Figure 3 . Performance of the biological attractor network. Shown are spike traces from the 8 pyramidal cells following twoentorhinal inputs at the onset of each theta cycle. Idealized gamma and theta population oscillations are shown for reference. Acomparison with Figure 2 shows that the correct attractors are reached by the end of the theta cycle, but that different trajectories aretaken than in the artificial attractor network. The hexadecimal equivalents of the network inputs and outputs are shown at the top.

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DISCUSSION

The biological attractor network

It should be noted that the results presented here are of a preliminary nature and our primary goal wasto present a concept for biological attractor networks, describe the structure of such a network, anddemonstrate its practical feasibility. Nevertheless, use of only eight pyramidal cells, while facilitating thecomparison with an artificial attractor network as well as making practical the simulation on a local networkof workstations, is a serious limitation. Storing just two patterns heavily overloads the storage capacity ofsuch a small network. A much larger version of this network has since demonstrated the scalability of themodel13.

Another problem encountered by the network is the relatively frequent (approximately 40%) failure toconverge to the appropriate attractor in the time window provided by the theta rhythm. This flaw is due tothe 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 toany of several stored patterns even in the presence of significant noise13.

Acetylcholine and the functional role of spikes and bursts

Physiological recordings show that hippocampal pyramidal cells are capable of either bursting orspiking, and that these different modes are correlated with the behavioral state of the rat15. From atheoretical perspective, these two firing modes have their respective advantages. Spiking is rapid and canhave a temporal precision of a millisecond or so, allowing for efficient representation of information and a

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Figure 4. Muscarinic neuromodulation of a pyramidal cell. (Top) Shown are 2.5 seconds of simulated somatic recordings from themodel pyramidal cell at various levels of cholinergic input. The cell received 0.1 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) hippocampalpyramidal cells. Layers 1-3 are the basal dendrites, layer 4 is the soma, and layers 5-11 are the apical dendritic arbor.

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well-defined network activity state. However, our simulations have demonstrated that spiking may be poorlysuited for inducing and/or maintaining synaptic plasticity in the distal dendritic arbor. Rather, this functionappears to be better fulfilled by backpropagating bursts which are capable of causing significant alterationsof calcium levels in the dendrites. The drawback for hippocampal bursts lies in their typical low-frequencyand 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 theinitial processing of novel information and its later recall, while bursting is necessary for more permanentstorage of patterns via the induction of LTP and LTD. This view is wholly consistent with Buzsáki’s “two-stage” memory model16. Our model indicates that ACh acts on at least two levels to initiate and manage atransition from bursting to spiking behavior, at least in intrinsically bursting hippocampal pyramidal cells. Atthe cellular level, the transition is due to a reduction in the afterhyperpolarizing calcium-dependent potassiumcurrent and the high-threshold calcium current. At the network level, the diffuse depolarizing action of AChon interneurons can serve as the driving force for mutually inhibitory interneuronal networks which cangenerate 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 beta-test 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

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2. D.J. Amit, Modeling Brain Function: The world of attractor neural networks, Cambridge University Press, New York, 1989.

3. X.J. Wang and G. Buzsáki, Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model, Journal ofNeuroscience, 16:6402-13 (1996).

4. R.D. Traub, M.A. Whittington, I.M. Stanford and J.G.R. Jefferys, A mechanism for generation of long-range synchronous fastoscillations in the cortex, Nature, 383:621-4 (1996).

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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 pyramidalneurone, 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 ininhibitory cell, Journal of Computational Neuroscience, 2:291-8 (1995).

10. R.D. Traub, M.A. Whittington, S.B. Colling, G. Buzsáki and J.G.R. Jefferys, Analysis of gamma rhythms in the rathippocampus in vitro and in vivo, Journal of Physiology, 493:471-484 (1996).

11. D.V. Madison, B. Lancaster and R.A. Nicoll, Voltage clamp analysis of cholinergic action in the hippocampus, Journal ofNeuroscience, 7:733-41 (1987).

12. M. Toselli and H.D. Lux, GTP-binding proteins mediate acetylcholine inhibition of voltage dependent calcium channels inhippocampal neurons, Pflugers Archiv - European Journal of Physiology, 413:319-21 (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 voltage-clamp data on intrinsic conductances, Journal of Neurophysiology, 66:635-50 (1991).

15. R.D. Traub and R. Miles, Neuronal Networks of the Hippocampus, Cambridge University Press, New York, 1991.

16. G. Buzsáki, Two-stage model of memory trace formation: a role for "noisy" brain states, Neuroscience, 31:551-70 (1989).