Modeling hippocampal theta oscillation: Applications in neuropharmacology and robot navigation

Modeling Hippocampal Theta Oscillation:Applications in Neuropharmacologyand Robot NavigationTamás Kiss,1,* Gergo Orbán,1 Péter Érdi1,2

1Department of Biophysics, KFKI Research Institute for Particle and NuclearPhysics of the Hungarian Academy of Sciences, Budapest, Hungary2Center for Complex Systems Studies, Physics Department,Kalamazoo College, Kalamazoo, Michigan, USA

This article introduces a biologically realistic mathematical, computational model of theta~�5 Hz! rhythm generation in the hippocampal CA1 region and some of its possible furtherapplications in drug discovery and in robotic/computational models of navigation. The modelshown here uses the conductance-based description of nerve cells: Populations of basket cells,alveus/lacunosum-moleculare interneurons, and pyramidal cells are used to model the hippo-campal CA1 and a fast-spiking GABAergic interneuron population for modeling the septal influ-ence. Results of the model show that the septo-hippocampal feedback loop is capable of robusttheta rhythm generation due to proper timing of pyramidal cells and synchronization within thebasket cell network via recurrent connections. © 2006 Wiley Periodicals, Inc.

1. INTRODUCTION

It is generally agreed that the hippocampal formation has a crucial role inlearning and memory processes.1 The hippocampus is reciprocally connected tomany neural centers and it is thought to prepare information for long-term storage.It has an important role in neurological diseases. Alzheimer’s disease, epilepsy,and ischemia are associated with learning and memory impairment and are accom-panied by selective neuronal death or characteristic changes in the hippocampalcircuitry. Recent studies have also indicated that the hippocampal formation isaffected in human depression as well as in animal models of depression2,3 andanxiety.4

*Author to whom all correspondence should be addressed: e-mail: [email protected].†e-mail: [email protected].‡e-mail: [email protected].

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 21, 903–917 ~2006!© 2006 Wiley Periodicals, Inc. Published online in Wiley InterScience~www.interscience.wiley.com!. • DOI 10.1002/int.20168

It is also known that the hippocampus plays an important role in spatial rep-resentation, orientation, and navigation5 in accordance with the cognitive maptheory.6

Two main, normally occurring, global hippocampal states are known: therhythmic slow activity, called the theta rhythm with the associated gamma oscil-lation, and the irregular sharp waves with the associated high-frequency ~ripple!oscillation. A pathological brain state, associated with epileptic seizures, is alsoknown to occur in the hippocampus.7

Based on experimental observations in rats, theta oscillation in the hippocam-pal formation is commonly regarded as one of the key components of the physio-logical basis of memory formation.8,9 Although most of our information onhippocampal oscillatory activity is derived from electrophysiological recordingsin anesthetized and nonanesthetized rodents, recent findings also indicate a simi-lar oscillatory activity in the hippocampal formation of the human brain.10–12 Ingeneral, theta oscillation in the hippocampal system has been linked to mnemonicprocesses,11,13–15 and the power of the theta oscillation seems to correlate with theanxiety level.16–19

In this article we integrate several aspects of hippocampal theta rhythms. First,normal and pathological rhythms are briefly reviewed from the perspective of non-linear dynamics. Then we give a mathematical model for the generation and con-trol of hippocampal theta rhythms. The conductance-based model presented consistsof four populations of neurons. Pharmacological agents could modify both mentaland behavioral states and their electrophysiological correlates. We offer a newcombined physiological/computational approach to drug discovery by finding opti-mal temporal patterns. We also suggest that our biologically realistic algorithmsfor the generation of theta rhythms should be incorporated into a higher-level, stillbioinspired navigating algorithm.

2. NORMAL AND PATHOLOGICAL BRAIN RHYTHMS

From EEG studies it is known that there are some well-defined rhythmic brainactivities underlying behavioral phenomena ~see Table I for a summary!. For exam-ple it was observed that in the hippocampal formation, learning and the composi-tion of memory traces are accompanied by a series of specific oscillations. Theso-called theta rhythm is characterized by a 1 mV amplitude, 4–12 Hz frequency

Table I. Characterization of normal and pathological hippocampal rhythms based on theiramplitude, duration, and frequency.7

Ripple

Name ENO SPW u b g Slow Fast

Amplitude ~mV! 17–25 3.5 1 �1 �1 �1 �1Duration ~ms! 190–750 10–120 15–50 40–120 40–120Frequency ~Hz! 0.003–0.06 0.2–5 4–12 10–25 20–80 100–140 140–200

904 KISS, ORBÁN, AND ÉRDI

International Journal of Intelligent Systems DOI 10.1002/int

periodic electrical activity. It is observed while the animal is in the REM sleepperiod or during walking or while being engaged in sensory scanning or explora-tion. Theta activity was found to often co-occur with another well-known rhythm,the gamma oscillation characterized by approximately 50 mV amplitude and40–100 Hz frequency. Both are population phenomena with correlated single-cellactivity. Theta activity is thought to be involved in time-locking cell activities,time stamping for phase coding, increasing the signal-to-noise ratio in neural cir-cuitries, and regulating cellular learning. Gamma activity plays a role in the bind-ing of perceived and recalled attributes of aspects and events. The two activitiestogether could be enrolled in the formation of memory traces and in sequencerecall. The mechanism underlying the generation of these brain activities and,accordingly, the relevant neural architecture is debated: Theta activity might bedelivered to the hippocampus from the septum by feed-forward inhibition or, con-versely, might be generated intrahippocampally. Gamma activity is supposed to begenerated locally in the hippocampus but could result from mutual excitation,mutual inhibition, or probably from an excitatory–inhibitory loop. Recent compu-tational studies suggest that a likely mechanism incorporating mutual inhibitionand delays in the input of cells might simultaneously account for the generation ofboth the theta and the gamma rhythms in the hippocampus.20

Epilepsy is a typical example of dynamical diseases.21 A dynamical diseaseoccurs in an intact physiological system due to the impairment of the control sys-tem, and is characterized by altered dynamic behavior. Epilepsy itself is character-ized by the occurrence of seizures ~i.e., ictal activities!. Epileptic seizures mightbe the product of combined cellular and network activities.22 During epileptic sei-zures, oscillatory activities emerge, which usually propagate through several dis-tinct brain regions. The epileptic neural activity is generally displayed in the localfield potential measured by local EEG. The epileptic activity occurs in a popula-tion of neurons when the membrane potentials of the neurons are “abnormally”synchronized. Both experiments and theoretical studies suggest the existence of ageneral synchronization mechanism in the hippocampal CA3 region. Synaptic inhi-bition regulates the spread of firing of pyramidal neurons. Inhibition may be reducedby applying drugs to block ~mostly! GABAA receptors. If inhibition falls below acritical level, complete synchrony occurs. Rather arbitrarily, activity has been con-sidered epileptic if more than 25% of the cells fire during 100 ms. In vitro modelsof epilepsy offer a means to study the cellular mechanisms of the different types ofepileptic phenomena by combined physiological and simulation methods. Severalin vitro models of seizures have been developed, including electrical stimulation,low calcium, low magnesium, and elevated potassium levels. Dynamical systemtheory offers a conceptual and mathematical framework to study epileptogenesis.Analytical studies based on bifurcation theory should clarify the possible operat-ing modes of a given neural network ~Figure 1!.23 The balance between excitationand inhibition is certainly one important control parameter, and its change mayimply transition between the regimes. Epileptic activities may be considered aschaotic processes, but are probably of lower dimension than normal EEG. Therehas been some hope that techniques of controlling chaos may offer new therapeu-tic and diagnostic tools for controlling epileptic activities.24

MODELING HIPPOCAMPAL THETA OSCILLATION 905


3. GENERATION AND CONTROL OF THE HIPPOCAMPALu OSCILLATION

In previous sections some aspects of neural oscillations have been discussedin general. In this section, a specific oscillation, the hippocampal theta rhythm,will be examined. As mentioned in Section 2, the theta oscillation of differentbrain areas might account for various aspects of information processing such asencoding of cognitive maps, learning, and navigation. Studying the mechanism oftheta rhythm generation and its connection with the functional role of the hippo-campus could enhance our understanding on how animals can rapidly execute com-plex navigation tasks. Furthermore, this knowledge might be incorporable intocomputational models and algorithms used to simulate animal and robot naviga-tion ~see, e.g., Refs. 25 and 26!.

Besides its role in information processing, hippocampal theta oscillation wasfound to be in connection with mood and emotions and some associated illnessessuch as stress or anxiety.19,27 By modifying hippocampal theta oscillation by meansof proper pharmacological agents, the desired change in mood or emotions mightbe evoked. From a connectionist point of view, finding the proper drugs means thefinding of chemicals acting on neural sites specific enough to alter only the desiredfunctions and only in the desired manner. To further elaborate on this question, themechanism of hippocampal theta oscillation generation should be approached.

3.1. A Mathematical Model of Theta Generation inthe Hippocampal CA1 Region

A computational model was developed to realistically account for the gener-ation of hippocampal theta oscillation in the CA1 region.28 Five cell populationslikely to play a key role in theta activity generation are described in the modelextending the Hodgkin–Huxley formalism.29 In this general approach ionic cur-rent flow through the cell membrane is described by combining the Kirchoff equa-tions with a modified version of a first-order description of reaction kinetics:

Figure 1. Different modes of brain activity: from stable fixed point to chaotic behavior. Theupper trace shows EEG signals for different control parameter ~ratios of excitatory @E# and inhib-itory @I# processes! values; below are corresponding phase space trajectories. An increase in theE/I ratio results in a bifurcation from a fixed point state to a chaotic state. Figure is based onLopes da Silva et al.23



CM

dV~t !

dt� Iext~t, q~t !!� IM~V, t ! ~1a!

IM~V, t ! � INa~V, t !� IK~V, t !� IL~V ! ~1b!

IL~V ! � gl~V~t !� EL! ~1c!

Ix ~t ! � gx ~V~t !, t !~V~t !� Ex !, x � $Na, K% ~1d!

gx ~V~t !, t ! � Sgx z~V~t !, t !, x � $Na, K%, z � $m, h, n% ~1e!

dz~V~t !, t !

dt� a~V~t !!~1 � z~t !!� b~V~t !!z~t !, z � $m, h, n% ~1f !

In Equation 1a, the balance equation for the membrane potential change isgiven with CM being the membrane capacitance and Iext the external or appliedcurrents injected through an electrode, where q denotes the eventual voltage orconcentration dependence. IM stands for the ionic membrane currents set forth inEquation 1b: IL is the leakage current representing various nonspecified ionic cur-rents and enabling us to set the resting membrane potential of the model. INa andIK are the sodium and delayed rectifier potassium currents, respectively. Maximalconductances of specific ions are given by the g~{! functions, further unfolded inEquation 1e. In this equation Sg is the maximal conductance, which describes spa-tial density and in a multicompartmental case the spatial distribution of a certainion channel. The m, h, n gating variables obey first-order kinetics according toEquation 1f.

Simulated neurons were hippocampal pyramidal cells, two types of horizon-tal interneurons: alveus/lacunosum-moleculare interneurons ~O-LM!, and sep-tally projecting horizontal cells, basket interneurons and septal GABAergic neurons~Figure 2!. The pyramidal cell model was derived from a previous model30 andwas extended with the hyperpolarization-activated current ~Ih!.32 The cell con-tained sodium ~INa!, delayed rectifier potassium ~IK!, A-type potassium ~IK~A!!,muscarinic potassium ~IK~M!!, C-type potassium ~IK~C!!, low-threshold calcium~ICa!, and calcium-concentration-dependent potassium ~IK~AHP!! currents. The Ih

current was described by equations of the standard Hodgkin–Huxley formalism:Ih � gh h~V � Eh!, where gh was the maximal synaptic conductance and Eh was thereversal potential of the current. Maximal conductance of Ih was 10 pS/cm2 at thesoma and increased linearly as a function of distance from the soma reaching amaximum at the most distal apical dendrites of 100 pS/cm2. The reversal potentialwas set to 0 mV. The gating variable h was described by first-order kinetics, in theform of dh/dt � ~hinf ~V ! � h!/th~V !. Values for hinf ~V ! and th~V ! were calcu-lated from an interpolated table ~Table II!.

Basket neurons formed the fast spiking neuron population of the pyramidallayer, containing INa and IK currents. These model neurons were previously usedin Refs. 20 and 33 to account for the population of fast, regularly spiking neurons.

The two types of horizontal neurons represented those interneuron popula-tions whose somata resided at the oriens/alveus border.34 These neurons weredescribed by the same set of equations, as their observed physiological properties



are similar and contained sodium, potassium, a high-threshold calcium, andhyperpolarization-activated currents.35 The basket and O-LM neurons were ableto generate repetitive action potentials autonomously, and O-LM neurons showedadaptation and low-frequency autonomous firing in the theta band.

Medial septal GABAergic neurons were single-compartment models previ-ously described in Ref. 34. This cell type evokes action potentials repeatedly inclusters. Between any two clusters the cell exhibits subthreshold oscillation but noaction potentials due to a slowly inactivating potassium current, which was added

Figure 2. ~a! Computer model of the hippocampal CA1 circuitry. Neuron populationshypothesized to be responsible for the generation of theta oscillation are shown ~pyr: pyramidalcells; i~O-LM!: horizontal cells projecting to the distal dentrites of pyramidal cells in thelacunosum moleculare layer; i~b!: basket interneurons; i~S!: septally projecting hippocampalhorizontal interneurons; MS-GABA: septal GABAergic cells, triangles denote excitatory, dotsinhibitory synapses!. Connections originating and ending at the same population denoterecurrent innervation. ~b! Morphology of the 265 compartmental pyramidal cell model ~seeRef. 30; the model described in the paper is available online as GENESIS31 code fromhttp://geza.kzoo.edu/theta/theta.html!.

Table II. Voltage dependency of the time constant th and steady-state level hinf of the gatingvariable h in Ih.

Vm

~mV!th

~ms! hinf

Vm

~mV!th

~ms! hinf

Vm

~mV!th

~ms! hinf

�140 17 1.0 �90.36 30 0.90 �52.72 20 0.063�129.09 17 1.0 �85.45 39 0.72 �41.81 16 0.040�118.18 20 0.98 �74.54 47 0.40 �30.90 11 0.0�107.27 24 0.96 �63.63 40 0.15 �20.00 8 0.0



to this model neuron besides the Hodgkin–Huxley type sodium and potassiumcurrents.

Connections within and among cell populations were created faithfully fol-lowing the hippocampal structure. The main excitatory input to horizontal neuronsis provided by the pyramidal cells via AMPA mediated synapses.36 Synapses ofthe septally projecting horizontal cells37 and synapses of the O-LM cell popula-tion innervating distal apical dendrites of pyramidal cells38 are of the GABAA

type. O-LM neurons also innervate parvalbumin containing basket neurons.39 Bas-ket neurons innervate pyramidal cells at their somatic region and other basket neu-rons40 as well. Septal GABAergic cells innervate other septal GABAergic cellsand hippocampal interneurons41,42 ~Figure 2!.

To characterize the population activity, a synthetic EEG was calculated. Asprincipal cells of the hippocampal CA1 region are pyramidal cells, for the calcu-lation of the local field potential only pyramidal cells were taken into account.These cells were placed on a circle of 400 mm radius with the recording electrodeat the center. The equation

F �1

4ps (i�1

n IMi

Ri

~2!

was used, where F is the field potential, s is the conductivity, IMiis the transmem-

brane current across the ith neural compartment, and Ri is the distance from theith neural compartment to the recording electrode.43

A detailed description of the model, parameter values, and GENESIS31 scriptfiles for computer simulations can be downloaded from http://geza.kzoo.edu/theta/theta.html. Figures shown were created using parameter values denoted as defaultin the above URL.

3.2. Generation of Theta Oscillation

This detailed, realistic model was used to examine the generation and controlof theta oscillation in the hippocampal CA1 region. As shown in Figure 3a, firingof neurons of the four populations was not evenly distributed in time, but timeintervals in which firing was significantly reduced were alternated by intervalswhere enhanced firing was observed. This synchronized state of neural firing wasfurther confirmed by the field potential, which exhibited a prominent �5 Hz oscil-lation as reflected in the power spectrum ~Figure 3b!.

Simulation results showed that key components in the regulation of the pop-ulation theta frequency are membrane potential oscillaton frequency of pyrami-dal cells, strength of pyramidal cell–O-LM cell innervation, and strength ofrecurrent basket cell connections. Membrane potential oscillation of pyramidalcells is determined by its average depolarization, passive membrane parameters,and parameters of the active currents. Average depolarization in our model resultsfrom septal cholinergic innervation. An important factor is the presence and max-imal conductance of the hyperpolarization activated current. If Ih is present, itshortens response times of pyramidal cells to hyperpolarizing current pulses and,



more importantly, decreases its variance: Ih acts as a frequency stabilizer. Syn-aptic strengths in our description are set by convergence numbers and maximalsynaptic conductances.

An explanation of intrahippocampal theta oscillation generation—based onthis model—includes ~1! signal propagation in the pyramidal cellrO-LM cellrbasket cellr pyramidal cell feedback loop; ~2! synchronization of neural activityvia the recurrent, inhibitory GABAA connections within the basket cell network;and ~3! synchronization of pyramidal cell firing due to rebound action potentialgeneration. It also worth noting that the propagation of a single signal throughoutthis trisynaptic loop would not require the amount of time characteristic of thetheta oscillation ~�0.2–0.25 s!; thus in the present case the population oscillationis created not by the propagation of single signals but rather the propagation of a“synchronized state” in the network. The observed periodic population activity isbrought about by alternating synchronization and desynchronization of cell activ-ities due to the interplay of the above mentioned synchronizing forces and somedesynchronizing forces ~such as heterogeneity of cell parameters and diversity ofsynaptic connections!, as observed in previous works.21,44

Figure 3. Appearance of theta frequency population activity in the firing of cells and theFourier spectrum of the field potential. ~a! Firing histograms were calculated by binning firingsof all cells of one of the four populations ~pyr: pyramidal cells, i~b!: basket cells, i~O-LM!:oriens-lacunosum moleculare interneurons, MS-GABA: septal GABAergic cells! into discretebins. The resulting graph shows the total activity of the respective population. ~b! Power spectrumof the field potential calculated by Equation 2. Theta frequency population activity is reflectedby temporal modulation of firings in ~a! and the �5 Hz peak in the power spectrum ~b!.



3.3. Pharmacological Modulation of Theta Oscillation

From a pharmacological point of view, it is desirable to identify a specificsite of the network where modifications are effective in influencing the populationactivity. To achieve this, GABAergic synapses located at different cell populationswere changed independent of each other ~Figure 5, below! to simulate the effect ofallosteric GABAA receptor modulators. Here synapses were described similarly tothe Hodgkin–Huxley formalism:

Isyn � Sgsyn s~V � Esyn! ~3a!

ds

dt� aF~Vpre!~1 � s!� bs ~3b!

F~Vpre! �1

1 � exp�Vpre �Qsyn

K� ~3c!

with Isyn being the synaptic current, Sgsyn the maximal synaptic conductance, s thegating variable of the synaptic channel, Esyn the synaptic reversal potential, F~{!an activation function, a and b rate functions describing opening and closing ofthe gate of the synaptic channel, and Qsyn a threshold. In this simple scheme ~Fig-ure 4! we chose Sgsyn to be the parameter under control of pharmaceutical agentssuch as diazepam or FG-7142.28 For a more detailed description of drug action, a

Figure 4. Modeling the effects of allosteric GABAA receptor modulators. In a simple descrip-tion of synaptic transfer, the strength of synapses was modulated via the Sgsyn parameter in Equa-tion 3a in a dose dependent manner. Inset: Modeled inhibitory postsynaptic potentials before~smaller amplitude! and after ~larger amplitude! administration of positive GABAA allostericmodulator.



more sophisticated model of the synapse, including the description of the compo-sition and decomposition of intermediate chemicals, would be necessary, which ina large network model would require huge computational power.

Simulations of drug action revealed that increasing the synaptic conductanceinduced a shift from theta-periodic to non-theta-periodic behavior in a number ofseptal GABAergic neurons in correspondence with observations of in vivo exper-iments using allosteric GABAA modulators like diazepam and FG-7142.28 Thisobservation shows that the level of perturbation arising form inhibitory post-synaptic potentials of afferent GABA neurons is a crucial factor contributing totheta-periodic characteristics of neuronal activity.

It was also found that different GABAergic connections in the hippocampushave different roles in synchronizing neuron populations. Thus, we showed thatstrength of interconnections between basket neurons had a great influence on septo-hippocampal activity. Similarly, simultaneous modulation of connections betweenseptal and septal neurons together with the modulation of synaptic transmissionbetween basket and basket neurons also induced strong septo-hippocampal activ-ity. In contrast, selective modulation of the GABAA connection between septaland septal neurons alone failed to alter theta activity of the septo-hippocampalsystem ~Figure 5!, although it had a clear effect on activity of individual septalneurons. The existence of such phenomena is of crucial importance from a phar-macological point of view: It shows that the modulation of specific pathways of alocal neural circuitry might lead to different global behaviors. Specific drugs withfew side effects might be possible to design based on the above observation.

Figure 5. Site-specific modulations of GABAergic transmission in the septo-hippocampalmodel. Reducing synaptic strength to 50% of the control situation at different pathways selec-tively. Different pathways have a different effect on theta activity generation as reflected by thepeak amplitude in the theta band of the simulated field potential. Basket-to-basket cell con-nections play a key role in synchronizing population activity ~control: control situation; totalreduction: reducing synaptic strength at all pathways; S2m, m2S: septally projecting horizontal-cell-to-septal-cell connections and vice versa; m2m: septal-to-septal recurrent connection;b2b: basket-to-basket recurrent connection!.



4. SPECULATIONS ON POSSIBLE APPLICATIONS OF THE MODEL

4.1. Toward a Computational Neuropharmacology

Computer models, like the one introduced above, have proven to be bene-ficial in the understanding of basic phenomena taking place at different structuraland functional levels. Mathematical models can help in arranging already exist-ing information and, through synthesis, can forecast never-seen properties andphenomena.

We have shown that in a moderately complex conductance-based model ofthe hippocampal CA1 region, theta rhythm generation can be observed and majorinteractions between cell populations and within cells responsible for the phenom-ena can be identified. These results qualify the model for consideration as a usefultool in the hands of computational neuroscientists, pharmacologists, and biolo-gists to complete their repertoire of available tools in the search for efficient andspecific drugs.

Briefly, we suggest the following method ~see Figure 6 for a graphical expla-nation!: First, connection between electrical brain activity and behavior or moodis identified,19,27 yielding the so-called desired pattern. In the presented case, thispattern consists of firing rate and timing of firings of all modeled cell populationsas well as the local field potential measured in the hippocampal CA1 region. Indeed,the identification of such a pattern is a highly nontrivial task, as, from a behavioralpoint of view, a certain pattern might correspond to multiple cognitive functionsor moods. In the case of theta rhythms, it is known that, on one hand, anxiolyticdrugs reduce hippocampal theta frequency4,18 by impairing the subcortical con-trol of hippocampal theta activity, whereas, on the other hand, hippocampal thetarhythm is essential for the animal in different memory and cognitive tasks.45,46

The power of theta and alpha oscillations in humans reflect cognitive and memory

Figure 6. Computational neuropharmacology—an idealized method for drug discovery. Seetext for details.



performance47 in a nontrivial way. Thus, to achieve good results, the desired pat-tern has to be detailed, containing all available information.

Second, a mathematical model is constructed, which—when its output isinterpreted—gives a modeled pattern comparable with the desired pattern. Mea-sures to carry this comparison out have to be set up in such a way that an auto-matic parameter-space search can be performed based on the quantitative andqualitative results of the comparison.

Last, when the modeled pattern sufficiently matches the desired pattern, modelparameters have to be read out and interpreted. In the presented case of hippocam-pal theta oscillation modulation by GABAA allosteric modulators, the importantparameters were the Sgsyn, synaptic strength parameters between different cell pop-ulations. As a result of this modeling, connections effective in the modulation oftheta rhythm were found.

We offer a new combined physiological/computational approach to drug dis-covery by finding optimal temporal patterns. Using modeling results, the drugscreening phase of the drug discovery process can be made more effective by nar-rowing the test set of possible target drugs. Besides this benefit, more selectivedrugs can be designed if the modeling method is able to identify some specificsites of drug action.

4.2. Theta Generation and Navigation

Hippocampal theta oscillation was found to be involved in navigation tasksand the identification of locations of the animal. Place cells, neurons that fire pref-erentially when the animal is in the vicinity of a certain place ~place field!48 arethought to establish a cognitive map in different hippocampal areas. These neu-rons use a double coding strategy of the position: on one hand, when the animalcrosses the place field of a place cell, the firing rate of the place cell is increased,creating a rate coding of the location. On the other hand, phase of pyramidal cellfiring relative to local field potential theta rhythm creates a phase code via thephase precession phenomena.49 Several mathematical models were proposed todescribe pyramidal cell activity during location coding.50–54

In recent papers,25,26,55 an abstract but biologically motivated model of nav-igation was presented. The hierarchically built model of the hippocampal forma-tion and the sensory cortex is designed as a KIV model that is built up of lowerlevel K sets, which—when coupled together to form the KIV set—exhibits cha-otic behavior. The model implements chaotic principles in a computer environ-ment and uses chaotic attractors to store memory traces in them.26,56 Chaos wasfound to be useful in a navigating algorithm,57 and chaotic processes were shownto be good candidates of encoding information in different brain structures.58– 60

Experimenting with the model showed that learning of environmental clues in anaperiodic ~chaotic! dynamical system requires theta framing of sensory informa-tion, similarly to that observed in behaving animals. This theta periodic samplingis achieved in the model by designing a learning cycle in which “active” periodssuch as the presentation of stimuli are followed by an approximately 100 ms rest-ing period.



By combining elements of the K sets ~starting from K0, an elementary build-ing block describing second-order dynamics of neural populations! constitutingthe hippocampal formation KIII set such that their emergent activity approachesthe synchronizing–desynchronizing population activity described in Section 3.2,besides conserving their resultant aperiodic character, the internally generated thetaactivity could be used to serve as a framing signal in the model.

5. CONCLUSIONS

In this article, a novel model of hippocampal theta rhythm generation wasproposed. The model uses techniques of conductance-based modeling, implement-ing four distinct, biologically realistic neuron populations. As a result of the model,a mechanism for rhythm generation was proposed. By exploiting the detailed con-nection structure of the model, selectively modifying synaptic strength param-eters, predictions were given for an effective method for hippocampal thetaoscillation modification by pharmacological means. Furthermore, the model givesa general scheme of long-period rhythm generation that can be incorporated intodifferent abstract modeling environments.

Acknowledgments

This work was supported by Hungarian Scientific Research Fund ~OTKA!Grant T-038140, the Henry Luce Foundation, and the Pfizer Corporation. Thanksto Robert Kozma for motivation.

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Modeling hippocampal theta oscillation: Applications in neuropharmacology and robot navigation