Modeling the Nonlinear Dynamic Interactions of Afferent Pathways

in the Dentate Gyrus of the Hippocampus

ANGELIKA DIMOKA,1 SPIROS H. COURELLIS,2 VASILIS Z. MARMARELIS,2 and THEODORE W. BERGER2,3

1247A Bourns Hall, Department of Bioengineering, Bourns College of Engineering, University of California, Riverside,Riverside, CA 92521, USA; 2Department of Biomedical Engineering, Viterbi School of Engineering, University of SouthernCalifornia, Los Angeles, CA 90089-1111, USA; and 3Neuroscience Program, University of Southern California, Los Angeles,

CA 90089-2520, USA

(Received 23 October 2007; accepted 6 February 2008; published online 26 February 2008)

Abstract—The dentate gyrus is the first region of thehippocampus that receives and integrates sensory informa-tion (e.g., visual, auditory, and olfactory) via the perforantpath, which is composed of two distinct neuronal pathways:the Lateral Perforant Path (LPP) and the Medial PerforantPath (MPP). This paper examines the nonlinear dynamicinteractions among arbitrary stimulation patterns at thesetwo afferent pathways and their combined effect on theresulting response of the granule cells at the dentate gyrus.We employ non-parametric Poisson–Volterra models thatserve as canonical quantitative descriptors of the nonlineardynamic transformations of the neuronal signals propagatingthrough these two neuronal pathways. These Poisson–Volterra models are estimated in the so-called ‘‘reducedform’’ with experimental data from in vitro hippocampalslices and provide excellent predictions of the electrophysi-ological activity of the granule cells in response to arbitrarystimulation patterns. The data are acquired through acustom-made multi-electrode-array system, which stimulatedsimultaneously the two pathways with random impulse trainsand recorded the neuronal postsynaptic activity at thegranule cell layer. The results of this study show thatsignificant nonlinear interactions exist between the LPP andthe MPP that may be critical for the integration of sensoryinformation performed by the dentate gyrus of the hippo-campus.

Keywords—Hippocampus, Dentate gyrus, Perforant

pathway, Poisson–Volterra, Kernel, Laguerre expansion,

Multi input, Multi electrode.

INTRODUCTION

The hippocampus receives and integrates neuro-nal activity from multiple brain regions that areinvolved in processing different modalities of sensoryinformation, especially in the context of learning and

memory.38,47 The primary function of the hippocam-pus is to form mnemonic labels that identify a unifiedcollection of features to create semantic and temporalrelations between multiple collections of features.6,24

The different modalities of sensory information prop-agate through synaptic connections to converge onto apopulation of common postsynaptic neurons. Thesesynaptic connections and the complex interactions ofthe sensory information are critical for the integrativefunctions of the hippocampus. However, due to thecomplexity and intrinsic nonlinearities of these inter-actions, the synaptic connections and the interactionsof the different types of sensory information have notbeen adequately studied in the literature. A fewresearchers have attempted to model the multi-inputinteractions in the central nervous system35,50 andother parts of the body,15,18,44 while others haveinvestigated the influence of prior synaptic activity onthe subsequent induction of synaptic plasticity in theform of long term potentiation and long term depres-sion in the dentate gyrus1,22,59 or CA1.34,58 In order toadvance our understanding of the mechanisms thatunderlie learning and memory, we seek biologicallyinterpretable models of how different modalities ofsensory information are processed and integrated inthe hippocampus.

Specifically, we focus on the functional interactionsof two pathways on the dentate gyrus that representsthe first integration level in the hippocampus (Fig. 1).The dentate gyrus receives its primary inputs from theentorhinal cortex via the perforant path, which consistsof two distinct synaptic inputs/pathways, the lateralperforant path (LPP) and the medial perforant path(MPP), which receive and integrate cognitive infor-mation (e.g., visual, auditory, and olfactory) fromother brain regions.56,61 It is possible to isolate the LPPand the MPP experimentally, because they are ana-tomically and functionally distinct (Fig. 1).30,31,53

Address correspondence to Angelika Dimoka, 247A Bourns Hall,

Department of Bioengineering, Bourns College of Engineering,

University of California, Riverside, Riverside, CA 92521, USA.

Electronic mail: dimoka@ucr.edu

Annals of Biomedical Engineering, Vol. 36, No. 5, May 2008 (� 2008) pp. 852–864

DOI: 10.1007/s10439-008-9463-6

0090-6964/08/0500-0852/0 � 2008 Biomedical Engineering Society

852

The LPP originates in the ventrolateral entorhinal areaand generates synapses on the outer one-third of themolecular layer. The MPP arises in the dorsomedialentorhinal cortex and synapses on the granule celldendrites in the middle one-third of the molecularlayer. The synaptic responses elicited in granule cellsby activation of these two distinct pathways exhibit anumber of different physiological2,46,45 and pharma-cological28,33,36 characteristics. The output of thedentate gyrus is subsequently processed in the hippo-campus through several sub-regions that form a closedloop (see Fig. 1). These sub-regions primarily consistof cascaded excitatory connections that are organizedroughly transverse to the longitudinal axis of the hip-pocampus, and they can be viewed as a set of inter-connected, parallel circuits. The significance of thisanatomical organization is that transverse slices(400 lm thick) of the dentate gyrus may be maintainedin vitro and preserve a substantial portion of theirintrinsic transverse circuitry.

Mathematical models have been developed on thebasis of physiological and biophysical principles tocharacterize the functional properties of corticalareas.32,55,60 These hypothesis-based models containseveral parameters that are biologically interpretableand must be measured or estimated from experimentaldata (whereby the term ‘‘parametric models’’). Theparametric models are not easy to scale-up in the caseof multiple interconnected neuronal units or when thefunctional complexity of the system increases, becausethe number of parameters required to represent suchfunctional complexity usually becomes too large fora workable model. To overcome this limitation,

non-parametric models based on the input–outputapproach were introduced in the so-called ‘‘reducedform’’ about 20 years ago to offer a comprehensivefunctional representation of neuronal networks in thehippocampus.9,10,14,49 More recently, non-parametricmodels have been used for modeling the input–outputrelationship in the dentate gyrus11,7 or in the CA1region25 of the hippocampus. A recently acceptedpaper presents non-parametric models of the separateeffects of LPP and MPP stimulation on the granule cellactivity but does not examine the nonlinear interac-tions of the two inputs and their combined effect ongranule cell activity.21 In the present paper, we employa ‘‘reduced form’’ non-parametric model based on thePoisson–Volterra approach8,9,41,42 to model the non-linear dynamic interactions of the simultaneous LPPand the MPP stimulation and the combined effect ofthe two inputs on the granule cell activity in thedentate gyrus of the hippocampus.

The Poisson–Volterra modeling approach is math-ematically rigorous and yields canonical models withexcellent predictive capabilities, even when the systemexhibits complex nonlinear dynamics. The key featuresof these models are the Poisson–Volterra kernels, whichare quantitative descriptors of the system function interms of the input–output relationships and fully cap-ture all possible nonlinear interactions among theunderlying biological mechanisms.41,42 These kernelscan be estimated robustly from experimental input–output data and take the ‘‘reduced form’’ when theoutput population spike is synchronized with the inputimpulses.8,9,49 The non-parametric models are data-based (inductive), unlike the parametric models thatare hypothesis-based (deductive). In the context of thisstudy, the non-parametric model has an advantageover its parametric counterparts since it represents thecombined activities of all known and unknowncomponents of neuronal interactions without anyassumptions about the model structure and thenumbers or types of elements that may be contributingto the neuronal activity. Most importantly, the non-parametric modeling approach can be extended toencompass the increasing complexity of multipleinputs and outputs that is emerging with the recentavailability of multi-unit recordings using multi-electrode arrays.

In the present study, the LPP and MPP pathwayswere stimulated simultaneously with two independentsequences of impulses having a random (Poisson)distribution of inter-impulse intervals and the elicitedresponse of a population of granule cells—the outputof the neuronal system in this formulation—wasrecorded with the multi-electrode arrays that were usedto simultaneously stimulate the LPP and the MPP. Theemployed stimulus is termed the Random Impulse

FIGURE 1. Schematic of the hippocampal circuitry. Theperforant path of the dentate gyrus consists of two anatomi-cally and functionally distinct subdivisions: the lateral perfo-rant path (LPP) that arises in the ventrolateral entorhinal areaand synapses in the outer one-third of the molecular layer,and the medial perforant path (MPP) that arises in thedorsomedial entorhinal cortex and synapses to the granulecell dendrites in the middle one-third of the molecular layer.

Modeling the Nonlinear Dynamic Interactions in the Hippocampus 853

Train (RIT), and it has been shown to test the systemwith a variety of input patterns over a short experi-mental time in a manner that reveals all possibleinteractions within the system.8,9,21,27,41,43 The col-lected experimental data were used to estimate in re-duced form the Poisson–Volterra models for twoinputs that include up to third-order nonlinear inter-actions between the LPP and the MPP. The resultssuggest that third-order nonlinear interactions (quanti-fied by the cross-kernels in the model) must be includedin the reduced-form Poisson–Volterra model toachieve adequate predictive accuracy during co-acti-vation of the LPP and the MPP. The employedexperimental and computational methods are pre-sented in the following section. The results are pre-sented in the subsequent section, followed by theDiscussion section.

MATERIALS AND METHODS

Experimental Preparation

Thin slices were taken from the hippocampus ofadult Sprague-Dawley male rats, 7–9 months of ageusing standard procedures.21 Each slice was carefullypositioned on a multi-electrode array over an invertedmicroscope (DML, DMIRB, Leika, Germany). The

slice was positioned so that stimulating electrodescovered the inner blade of the dentate gyrus (Fig. 2a).Accurate positioning of the slice on top of the multi-electrode array was attained by floating the slice usinga very fine hair brush. The positioning of the slicerelative to the array was documented with a digitalcamera (Hitachi VK-C370, Spot Model 2.0.0).Throughout all experiments, slices were perfused witha 1 mM MgSO4 aCSF (artificial Cerebro-Spinal Fluid)at a flow rate of 15 mL/min, and constantly heated at33 �C. A multi-electrode array system was used tosimultaneously stimulate and record from multiplesites in each hippocampal slice. The system consistedof two components: (1) a multi-site electrode array,and (2) a multi-channel stimulation-recording system.The multi-site electrode array employed in this exper-iment is custom-designed and was fabricated andtested in the Center for Neural Engineering at theUniversity of Southern California.21,27 The array waspositioned so that its electrode sites cover the appro-priate sub-regions of the dentate gyrus (LPP and MPP)with sufficient density as to allow separate stimulationof each pathway. The multi-electrode array has 60microelectrodes in a 3 · 20 configuration. The micro-electrodes are embedded in a planar glass plate andtheir diameter is 28 lm with a center-to-center spacingof 50 lm. The multi-channel stimulation-recording

FIGURE 2. Selected electrodes for LPP and MPP stimulation (a). Characteristic field potential responses from the granule celllayer due to LPP stimulation (b) or MPP stimulation (c). The occurrence of the stimulation is denoted by ?: The negative-goingdeflection in each response is the population spike and its amplitude is measured as indicated by the vertical arrow. A segment ofRIT input impulses (with randomly varying inter-impulse intervals) of the two point-process stimuli applied to LPP (green) and MPP(red) along with the elicited population spikes at the granule cells (d).

DIMOKA et al.854

system is a commercially available product (MEA60Multi Channel Systems, Germany) that consists ofpreamplifiers, data acquisition card, 8-channel stimu-lation box and software for data acquisition and pre-processing (MC Stimulus v2.0.3, MC Rack v 2.2.2).

Each slice was positioned on top of the multi-elec-trode array as to cover the molecular layer from thefissure to the granule cell body layer at the upper bladeof the dentate gyrus. Electrodes positioned at the outerone-third of the molecular layer were chosen as can-didates to stimulate the LPP and at the middle one-third of the molecular layer to stimulate the MPP. Theselection of the optimum stimulating electrodes wasconfirmed by the following electrophysiological crite-ria: (1) field Excitatory Post-Synaptic Potentials(fEPSPs) showed paired-pulse facilitation when theLPP was stimulated, and paired-pulse depression whenthe MPP was stimulated;16,33,39,45 (2) fEPSPs exhibiteda dendritic current sink at the outer molecular layerand a dendritic current source at the middle molecularlayer when LPP was stimulated (accordingly, when theMPP was stimulated fEPSPs exhibited a dendriticcurrent sink at the middle molecular layer and a den-dritic current source at the outer molecular layer);16,20

(3) stimulation at MPP exhibited shorter latencies ofthe population spike recorded in the granule cell layerthan when the LPP was stimulated.8,10,45 The electrodepositioned right below the granule cell layer into thehilus was used to record the electrophysiologicalactivity of granule cells in the form of populationspikes that reflect the synchronous activation of den-tate granule cells.3,37

Experimental Protocol

At the beginning of each experiment, input–output(I/O) curves were recorded for the LPP and the MPPseparately. Each pathway was stimulated with a seriesof pulses having intensities that varied between 10 and140 lA, in 10 lA increments. Stimulation was deliv-ered in the form of bi-phasic pulses (100 ls duration)via the pair of stimulation electrodes appropriate foreach pathway. The time between successive pulses wasset to 30 s to avoid induction of long-term potentiation.Each series of pulses was repeated five times. The I/Ocurve for each pathway was used to determine the valueof the stimulation intensity that evoked 50% of themaximum population spike amplitude response in thegranule cell layer. This value was used for the remain-der of each experiment. Both pathways were simulta-neously stimulated with random impulse train (RIT)stimuli. Each RIT contained 800 Poisson-distributedimpulses (mean inter-impulse interval of 500 ms, cov-ering the approximate range of 10–4000 ms). Theparameter of the Poisson distribution that determines

the average firing rate of the RIT was consistent withthe known firing rates of the respective hippocampalneurons (2 Hz).51,57 At the end of RIT stimulation, thesystem was tested for stationarity by administering aseries of I/O curve stimulations. Possible differences inthe resulting I/O curves before and after RIT stimula-tion were taken as indicators of changes in granule cellexcitability (non-stationarity of the system). Only datathat exhibited levels of such changes within ±15% ofthe baseline were included in the analysis.

The experimental data were sampled at 25 kHz perchannel and the amplitude of each population spikewas extracted for analysis using a customized interfacewritten in Matlab (v6.5). The amplitude of the popu-lation spike was defined as the segment of the verticalline between the negative peak of the spike and thetangential straight line connecting the spike onset andoffset (see Figs. 2b and 2c).

Poisson–Volterra Modeling of a Two-Input System

The collected data were analyzed using a variant ofthe general Volterra modeling approach43,41,42 adaptedto the two-input case. The two inputs are independentPoisson RIT stimuli and the output is the resultingsequence of population spikes with variable amplitude.This approach considers the input and the outputspikes to be synchronized, i.e., to occur in the sametime-bin which is set equal to 10 ms so that it exceedsthe maximum output latency of 8.2 ms observed in ourexperiments.

The point-process inputs of the system (LPP andMPP) are RIT sequences of impulses expressed as:

xL ¼X

njL

dðn� njLÞ ð1Þ

xM ¼X

njM

dðn� njMÞ ð2Þ

where the discrete-time delta function (Kroneckerdelta) denotes an impulse at the indicated discretetime-index nj, and the subscripts L and M denote theLPP and MPP pathways, respectively. The populationspike amplitude that is generated in the granule celllayer in response to a stimulating impulse either at theLPP or the MPP is given by:

yðniÞ ¼ yLðniLÞ þ yMðniMÞ ð3Þ

where y(ni) is the output at discrete time ni, which is thetime of occurrence of the i-th stimulating impulseeither at the LPP or the MPP. An implicit assumptionis that stimulating impulses do not occur simulta-neously at both pathways, so that only one pathwaycan be stimulated at each time instant ni. In the

Modeling the Nonlinear Dynamic Interactions in the Hippocampus 855

adopted modeling formulation, the output is com-posed of two components corresponding to the timingof the impulses of each of the two inputs that are givenby the expressions of the reduced-form Poisson–Vol-terra model with two inputs as:

yLðniLÞ ¼ k1yL þX

njL

k2yLxLðniL � njLÞ

þX

njM

k2yLxMðniL � njMÞ

þX

njL

X

nj0L

k3yLxLxLðniL � njL ; niL � nj0LÞ

þX

njL

X

nj0M

k3yLxLxMðniL � njL ; niL � njMÞ

þX

njM

X

nj0M

k3yLxMxMðniL � njM ; niL � nj0MÞ ð4Þ

and

yMðniMÞ ¼ k1yM þX

njL

k2yMxLðniM � njLÞ

þX

njM

k2yMxMðni2 � nj2Þ

þX

njL

X

nj0L

k3yMxLxLðniM � njL ; niM � nj0LÞ

þX

njL

X

nj0M

k3yMxMxLðniM � njM ; niM � njLÞ

þX

njM

X

nj0M

k3yMxMxMðniM � njM ; niM � nj0MÞ ð5Þ

where the discrete-time index nj over which summationoccurs in the above expressions is the time of occur-rence of any prior j-th impulse within a time window lstarting at present and extending into the past (termedthe ‘‘system memory’’ and representing the input pastepoch exerting causal effects on the output presentvalue). Note that these prior impulses may occur ateither stimulating pathway (LPP or MPP), as indicatedby the respective subscript of the time index. Thus, astimulating impulse at one pathway may exert a causaleffect on the granule cells through either pathway overa subsequent time window equal to the system mem-ory l. The effects that are due to the same pathway arerepresented by the ‘‘self-kernels’’ and those due to theother pathway are represented by the ‘‘cross-kernels’’(see below). These two sets of reduced-form Poisson–Volterra kernels are clearly separated and constitutethe key features of this Poisson–Volterra model,42

since they describe the quantitative effects of the inputson the output and provide the model with its predictivecapability. The estimation of these kernels using actualinput–output data is the key objective of the modeling

task in this context. In the expressions (4) and (5)above, the following reduced-form Poisson–Volterrakernels are denoted by the proper subscript for therespective input (L for LPP or M for MPP): k1yL andk1yM are the first-order kernels (constants), k2yLxLðnÞand k2yMxMðnÞ are the second-order self-kernels (onedimensional), k2yLxMðnÞ and k2yMxLðnÞ are the second-order cross-kernels (one dimensional), k3yLxLxLn1; n2and k3yMxMxMðn1; n2Þ are the third-order self-kernels(two dimensional), and k3yLxLxMðn1; n2Þ; k3yLxMxMðn1; n2Þ;k3yMxMxLðn1; n2Þ and k3yMxLxLðn1; n2Þ are the third-ordercross-kernels (two dimensional). The obtained esti-mates of these kernels are shown in the followingsection.

The first-order kernel represents the amplitude of thepopulation spike attributed to the respective inputimpulsewhen it is acting alone—i.e., in the absence of anyother input impulse within the system memory. The sec-ond-order self-kernel represents the nonlinear dynamicinteractionswithina singlepathway (either theLPPor theMPP) between the present impulse and each of the pastimpulses of the same stimulus within the memorywindow l. Thesenonlineardynamic interactions result inpartial modulation of the output (the amplitude of thepopulation spike) in a manner dictated by the specificvalueof the second-order self-kernel corresponding to thespecific inter-impulse interval, in accordance with thereduced-form Poisson–Volterra model of Eqs. (3)–(5).Additional output modulation results from the otherterms of the model in a manner dictated by the respectivekernels. Thus, the second-order cross-kernel representsthe nonlinear dynamic interactions across two pathwaysfor the respective inter-impulse interval, i.e., between thepresent stimulus impulse in one pathway and each ofthepast stimulus impulses in theotherpathwaywithin thememory window l.

In analogous fashion, the third-order self-kernelrepresents the interactions between the present impulseand any two preceding impulses of the same stimuluswithin a single pathway over the memory window l,and the third-order cross-kernel represents the inter-actions between the present stimulus impulse in onepathway and any two preceding stimulus impulses inthe other pathway over the memory window l. Thespecific amount of third-order modulation of the out-put population spike is determined by the value of therespective third-order kernel at the corresponding twointer-impulse intervals. Therefore, knowledge of thevalues of these kernels provides complete quantitativeinformation about the nonlinear functional character-istics of the system (for the respective order of inter-actions), including the manner in which the twopathways interact to integrate incoming informationand produce the neuronal activity at the granule cellsof the dentate gyrus.

DIMOKA et al.856

In order to estimate efficiently and accurately thekernels from relatively short input–output data records,we approximate them with linear combinations ofexponentially decaying Laguerre functions through anorthogonal expansion which improves the estimationaccuracy and reduces the computational effort.40 TheLaguerre expansion coefficients are estimated usingleast-squares methods, and the kernel estimates arereconstructed using the respective estimated expansioncoefficients.17 The predictive accuracy of the obtainedmodel (employing the estimated kernels) can be evalu-ated using the Normalized Mean Square Error(NMSE) of the model prediction vs. the actual outputof the system, which is non-negative and defined as:

NMSE ¼P

i ðYpri � YdataiÞ2

PY2datai

i

ð6Þ

where Ypr is the predicted amplitude of the populationspikes using the estimated kernels and Ydata is the re-corded amplitude of the population spikes obtainedduring the experiment. The value of the NMSE is ta-ken as a measure of the predictive capability of themodel and becomes zero for a perfect prediction. Theinevitable presence of measurement errors and noise inthe data prevents the NMSE from becoming zero in apractical context, even if the model is perfect. There-fore, the interpretation of the numerical values of thecomputed NMSE must be made in the context of thenoise conditions and possible measurement errors ineach case. Typically, NMSE values of less than 0.2 aredeemed to indicate satisfactory predictive capability ofthe model that validates its specific form. Since thekernel estimation procedure employs Laguerre expan-sions, the NMSE value of the model predictiondepends on the selected number of Laguerre functionsfor kernel expansion (L) and the value of the Laguerreparameter a.41,42 The selection of the optimum valuesof a and L in each particular application is accom-plished through a search procedure over various valuesof L and a (varying in ascending order) that seeks tominimize the NMSE of the respective model predic-tion. The number L of Laguerre functions is selectedthrough the mentioned search procedure by requiringthat an increase of L by 1 causes reduction of theprediction NMSE by at least 3%.

The kernel estimation procedure and the nonlinearmodeling/analysis of this system were performed withthe use of a specialized software package (LYSIS) thathas been developed by the Biomedical SimulationsResource at theUniversity of SouthernCalifornia and isdistributed to the biomedical community free of charge.The statistical significance of the obtained estimatesof the Laguerre expansion coefficients (from whichthe kernel estimates and the Poisson–Volterra model

predictions are constructed) was evaluated by applyingthe Student’s t-test at the significance level of p<0.01.

RESULTS

Five experiments were performed in vitro using RITstimuli for the two inputs (LPP and MPP) and datarecords of 400 spikes per pathway for each experimentwere collected. Following a search procedure of suc-cessive trials and using the prediction NMSE as crite-rion for model performance, we selected the third-orderreduced-form Poisson–Volterra model described byEqs. (3)–(5) as the most appropriate for the analysis thedata collected in the RIT experiments. Three Laguerrefunctions (L = 3) were selected for the kernel expan-sions as a good compromise between prediction accu-racy and model complexity (since the total number offree parameters increases with L). For L = 3, the totalnumber of free parameters in the two-input model ofEqs. (4)–(5) is 62, which is less than a third ofthe number of output spikes and assures that we avoidthe risk of overfitting. For L = 3, the optimum valueof the Laguerre parameter a = 0.984 ± 0.004 wasselected as to minimize the NMSE of the modelprediction. Figure 3 shows the average NMSE derivedfrom the five different experiments for L = 3 and a

14

15

16

17

0.9 0.92 0.94 0.96 0.98 1

0.9 0.92 0.94 0.96 0.98 1

alpha

ES

MN

ES

MN

self kernel

5.5

6.5

7.5

alpha

self and cross kernel

FIGURE 3. The average NMSE values over five experimentsusing the third-order Poisson–Volterra model given byEqs. (3)–(5) for a range of a values from 0.90 to 0.99 and threeLaguerre functions (L = 3). For the optimum selection ofa = 0.984 and L = 3, the NMSE of the model prediction is14.39 ± 3.86% when only the self-kernels are included, anddrops to 5.88 ± 2.51% when both self-kernels and cross-kernels are included in the model.

Modeling the Nonlinear Dynamic Interactions in the Hippocampus 857

range of a values from 0.90 to 0.99. For these selectionsof L and a values, the NMSE was calculated both withand without including the cross- kernels in the third-order model. When only the self-kernels were included,the value of the NMSE was 14.39 ± 3.86%. Whenboth the self-kernels and the cross-kernels were in-cluded the NMSE dropped to 5.88 ± 2.51%. Theinclusion of the cross-kernels reduced the NMSE by8.52%. Application of the paired t-test indicated thatthe inclusion of the cross- kernels resulted in a statis-tically significant reduction of the NMSE (p<0.01),which implies that significant dynamic interactions ex-ist between the two pathways.

For each input–output dataset, the first-order, sec-ond-order and third-order kernels were estimated forL = 3 and the optimum value of the parametera = 0.984±0.002. Over the five experiments, theaverage value (±one standard deviation) of the first-order kernel (k1yLÞ for the LPP was 186.75 lV(±54.9 lV) and of the first-order kernel (k1yMÞ for theMPP was 276.65 lV (±22.1 lV). The average valuesand the standard-deviation bounds of the second-orderself-kernels and cross-kernels, normalized by the cor-responding first-order kernel value, are shown in Fig. 4

for the LPP and in Fig. 5 for the MPP. It is evident inFig. 4a that the average values of the normalized sec-ond-order LPP self- kernel k2yLxL exhibit initially a fastfacilitatory phase up to about 110 ms, followed by aslow inhibitory phase up to about 1600 ms. In Fig. 4b,the average values of the normalized second-ordercross-kernel k2yLxM exhibit initially a moderate facili-tatory phase that crosses into a very shallow inhibitoryphase around 200 ms and diminishes around 800 ms.By contrast, we observe in Fig. 5a that the averagevalues of the normalized second-order MPP self-kernelk2yMxM exhibit initially a very small facilitatory phaseup to about 70 ms, followed by a slow inhibitory phasethat diminishes around 1800 ms. In Fig. 5b, we seethat the average values of the normalized second-ordercross-kernel k2yMxL exhibit initially a moderate facili-tatory phase up to about 160 ms and cross into ashallow inhibitory phase that diminishes around1200 ms. We observe the distinct dynamic character-istics of the self-kernels for the two input pathways andthe similar dynamic characteristics of the cross-kernels.

The average values of the normalized third-orderself-kernels and cross-kernels (as well as their respec-tive standard-deviation bounds) are shown in Fig. 6

(b)

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 200 400 600 800 1000 1200 1400 1600 1800 2000

time (msec)

(a)

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 200 400 600 800 1000 1200 1400 1600 1800 2000

time (msec)

k dezilamron

2yLx L

k dezi lamro n

2yLx M

second order self kernel

second order cross kernel

LATERAL PERFORANT PATH

FIGURE 4. The average values of the normalized second-order self-kernel (k2y1x1

) and cross-kernel (k2y1x2) for the LPP.

The dotted line shows ±one standard deviation for five dif-ferent experiments. In (a), the average values of the normal-ized self-kernel exhibit initially a fast facilitatory phase up toabout 110 ms followed by a slow inhibitory phase up to about1800 ms. In (b), the average values of the normalized cross-kernel exhibit initially a moderate facilitatory phase thatcrosses into a very shallow inhibitory phase shortly around200 ms and diminishes around 800 ms.

MEDIAL PERFORANT PATH second order self kernels

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 200 400 600 800 1000 1200 1400 1600 1800 2000

time (msec)

2k dezila

mro

n

second order cross kernels

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 200 400 600 800 1000 1200 1400 1600 1800 2000

time (msec)

k dezilamron

2yM

x Mk dezila

mron2y

Mx L

(a)

(b)

FIGURE 5. The average values of the normalized second-order self-kernel (k2y2x2

) and cross-kernel (k2y2x1) for the MPP.

The dotted line shows ±one standard deviation for the fivedifferent experiments. In (a), the average values of the nor-malized self-kernel exhibit initially a very small facilitatoryphase up to about 70 ms, followed by a slow inhibitory phasethat diminishes around 1800 ms. In (b), the average values ofthe normalized cross-kernel k2y2x1

exhibit initially a moderatefacilitatory phase up to about 160 ms and cross into a shallowinhibitory phase that diminishes around 1200 ms.

DIMOKA et al.858

for the LPP and in Fig. 7 for the MPP. The averagevalues of the normalized third-order LPP self-kernelk3yLxLxL are shown in Fig. 6b and the respective stan-dard-deviation bounds are shown in Fig. 6a (upperbound—i.e., average plus one standard deviation) andFig. 6c (lower bound—i.e., average minus one stan-dard deviation). It is evident in Fig. 6b that this kernelexhibits initially a strong inhibitory region that crossesinto two weaker (symmetric) facilitatory regions after100 ms. The latter regions diminish around 1400 ms.The meaning of the kernel values in the initial inhibi-tory region is that the effect of a pair of impulsesoccurring within 100 ms prior to the present impulse isnegative on the amplitude of the population spikecaused by the present impulse. Conversely, the kernelvalues in the facilitatory regions indicate that the effectof a pair of impulses occurring between 100 and1400 ms prior to the present impulse is positive on the

amplitude of the population spike caused by thepresent impulse. The size of this positive or negativeeffect is determined by the value of the kernel at thecoordinates that correspond to the two time-lags of thepair of ‘‘conditioning’’ (i.e., preceding) impulses (i.e.,the time between the present output spike and each ofthe ‘‘conditioning’’ input impulses).

The average values of the normalized third-orderLPP cross-kernels k3yLxLxM and k3yLxMxM are shown inpanels E and H, respectively. The correspondingstandard-deviation upper bounds are shown in panelsD and H, while the lower bounds are shown in panelsG and I, respectively. It is evident in Fig. 6e that thecross-kernel k3yLxLxM exhibits characteristics similarto the self-kernel in panel B with smaller facilitatoryregions. By contrast, we see in Fig. 6h that the cross-kernel k3yLxMxM exhibits opposite characteristics (i.e.,an initial strong facilitatory region that crosses into

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Modeling the Nonlinear Dynamic Interactions in the Hippocampus 859

two weaker inhibitory regions around 200 ms). Notealso that the cross-kernels are not symmetric in general(unlike the self-kernels that are symmetric by defini-tion).

In an analogous manner to the LPP case, the third-order kernels (self- and cross-) in the MPP case arepresented in Fig. 7. The average values of the nor-malized third-order MPP self-kernel are shown inFig. 7b and the respective standard-deviation boundsare shown in Fig. 7a (upper bound) and Fig. 7c (lowerbound). It is evident in Fig. 7b that this kernel exhibitsa similar morphology to its LPP counterpart. The sameobservation holds for the average values of the nor-malized third order MPP cross-kernels shown inFigs. 7e and 7h, respectively. The correspondingstandard-deviation upper bounds are shown in panelsD and H while the lower bounds are shown in panels G

and I, respectively. It is noted that the magnitudes ofthe third-order kernels are not similar for the twopathways. Specifically, the magnitude of the MPP self-kernel is larger its LPP counterpart, but this relation isreversed for the cross-kernels of the two pathways.

Predictive Capabilities of the Model

In addition to providing a quantitative descriptionof the nonlinear dynamic characteristics of the systemin the form of kernels, the proposed third-orderPoisson–Volterra model has predictive capabilities forarbitrary inputs. In Fig. 8, we show the actual outputdata (i.e., the sequence of population spikes inresponse to independent RIT stimulation at bothpathways) and the response predicted by the third-order Poisson–Volterra model for one representative

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experiment. Two ways of calculation for the model-predicted response are shown: one considering only theself-kernels and the other considering both the self-kernels and cross-kernels. It is evident that the pre-diction of the output is better when the cross-kernelsare included—a result that suggests that the interac-tions between the two pathways (captured by the cross-kernels) are significant for this system and must beincluded in the model. The predictive power of thePoisson–Volterra model can be further validated usingout-of-sample predictions, where the system output ispredicted for an arbitrary stimulus using kernels ob-tained from a different input–output dataset (segmentshown in Fig. 9). Visual assessment and the computedNMSE values demonstrate the out-of-sample predic-tive power of the Poisson–Volterra model. For therepresentative experiment in Fig. 8, the in-sampleNMSE is 5.21% and the out-of-sample NMSE is6.62%.

We should note that, if a second-order Poisson–Volterra model is used (i.e., a model including onlyfirst and second order kernels), then the optimuma = 0.992 and the model prediction for L = 3 hasNMSE values of 10.79 ± 4.48% (when including bothself- and cross-kernels). This demonstrates the superi-

ority of the third-order Poisson–Volterra model overthe second-order one, since the former yields predic-tion NMSE values of 5.88 ± 2.51% (t = 2.14,p<0.05, df = 7), (a statistically significant reduction).

DISCUSSION

Biologically interpretable models of the hippocam-pus that capture accurately its functional propertiesand the dynamic interactions among its various neu-ronal inputs are essential in order to understand howdistinct sensory modalities are processed and inte-grated within the hippocampus. In this study, we useexperimental data collected under random stimulationconditions to obtain and validate a non-parametric(Poisson–Volterra) model that describes accurately theeffect of the nonlinear dynamic interactions betweencombined stimulation at the LPP and the MPP path-ways on the granule cell activity in the dentate gyrus ofthe hippocampus. In the presented approach, thefunctional properties of this hippocampal neuronalnetwork are described reliably and quantitatively bykernel functions that are characteristic of the systemand define uniquely its input–output relation so thatthe output of the system can be accurately predicted inresponse to arbitrary inputs. These kernel functions aredetermined experimentally through a novel estimationprocedure utilizing data that are collected by stimu-lating two afferent pathways of the neuronal network(i.e., the LPP and MPP) with random (Poisson) se-quences of impulses to generate a broad repertoire ofinteractions among the network elements. Theseexperimental conditions allow us to observe theactivity of the system under a broad variety of oper-ating conditions (over the relatively short experimen-tation time) that are subsequently analyzed in thegeneral Poisson–Volterra modeling framework to giveus the capability of predicting the output for arbitraryinputs (generalization property). The kernels of theobtained Poisson–Volterra model represent a set ofcanonical descriptors of the functional characteristicsof the system and, as such, they can provide somevaluable insight into the neuronal mechanisms of thissystem. However, the non-parametric nature of themodel limits its ability to draw a definitive relationwith the underlying neuro-molecular structures ormechanisms.

These results demonstrate that the non-parametric(Poisson–Volterra) model captures the nonlineardynamic interactions between the LPP and the MPPand provides accurate prediction of the system outputto arbitrary inputs. It was shown that the inclusion ofthird-order kernels significantly enhances the predictivecapabilities of the model and, therefore, the actual

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Modeling the Nonlinear Dynamic Interactions in the Hippocampus 861

system possesses significant third-order nonlinearities(i.e., dynamic interactions among triplets of impulsesaffect the output) that are ignored in the widely-usedapproach of paired-pulse stimulation and associatedanalysis. Specifically, the value of the predictionNMSE dropped by 4.91% (t = 2.14, p<0.05, df = 7)when the third-order model was used instead of thesecond-order model. In addition, the inclusion of thecross-kernels in the third-order model reduced theprediction NMSE by 8.52% (from 14.39% to 5.88%,t = 4.13, p<0.01, df = 7) relative to the respectiveNMSE value when only the self-kernels were used inthe model (not including the pathway interactions)(Table 1). Therefore, the appropriate Poisson–Volterramodel that describes the nonlinear dynamic interac-tions of the two pathways of the perforant path on thegranule cell activity of the dentate gyrus is of third-order and must include both the self-kernels and thecross-kernels.

The performance of the third-order model wasdeemed satisfactory. Although we cannot rule out thepossibility of higher order terms, the amount ofavailable data cannot practically support an extensionto fourth-order models at this time (without risk ofover-fitting). Higher order models will be explored inthe future with the use of Principal Dynamic Modes.41

Although the application of the RIT stimulationand the analysis of the resulting data in the context ofPoisson–Volterra modeling requires greater computa-tional effort, this incremental computational effort isjustified by the valuable information contained withinthe obtained results. This brings us to the perennialquestion of the physiological significance and theinterpretation of the obtained Poisson–Volterra ker-nels in a manner that advances our scientific under-standing of the system. Consider, for instance, theearly depressive characteristics of the granule cells thathave been attributed to GABA-mediated IPSPs48 andGABA-mediated increase in chlorium conductancethat reduces the excitability of granule cells23 (for lagsless than 40 ms) or have been attributed to the recur-rent (feedback) activation of the GABAergic basketcells5,10,37 (up to lags of 100 ms). Such early negativevalues (representing depressive effects) are not seen inthe obtained second-order kernels but they are evidentin the third-order kernels (always in the self-kernelsand in one of the two cross-kernels). Our interpretation

is that the observed early depression is due to third-order effects in the presence of dual-pathway stimula-tion (LPP and MPP), while the previously reportedobservations were for single-pathway stimulation (LPPor MPP).21 This intriguing result indicates the impor-tance of studying the system of interest under condi-tions that resemble its normal operation (e.g.,combined and random stimulation), otherwise the re-sults may be partial and potentially misleading. Notethat the positive values exhibited in the early lags (upto 100 ms) of the second-order kernels may be attrib-uted to NMDA-mediated synaptic events. Similarfacilitation characteristics are observed at intermediatelags of the third-order LPP kernel and may be attrib-uted to augmentation of excitatory transmitterrelease4,19 or presynaptic inhibition of GABArelease.12,29 This is consistent with the reported facili-tation in the region of intermediate lags (100–200 ms).At longer lags (200–1000 ms), the obtained second-order kernels have negative values in agreement withthe depressive characteristics of granule cells reportedpreviously10,13 which may be due to a voltage-depen-dent and/or calcium-activated potassium conduc-tance.52,54 These effects are quantitatively reflected onthe estimated kernel values and separated into second-order and third-order effects. This initial interpretationof the obtained kernels is only a first step in a long-lasting effort that will be required before completephysiological interpretation can be achieved and thefull scientific benefit of this analysis can be realized.

The proposed approach offers the appropriatemethodological means for analyzing quantitatively the(nonlinear) dynamic interactions between any two (ormore) pathways of the hippocampus or other neuronalsystems. Following this approach, future research canexpand the mathematical model to include three ormore inputs of the dentate gyrus and to incorporateother hippocampal regions. Such a representation ofthe hippocampus can provide a compact quantitativerepresentation of the interactions among hippocampalpathways (which have been so far studied individually)that may advance our understanding of the complexneuronal mechanisms that integrate sensory informa-tion in the hippocampus. This remains a great chal-lenge for future research.

ACKNOWLEDGMENTS

This work was supported in part by NIH Grant No.P41 EB001978 to the USC Biomedical SimulationsResource and NSF Grant No. EEC-0310723 to theUSC Engineering Research Center on BiomimeticMicroElectronics Systems.

TABLE 1. NMSE comparison for the in-sample second andthird order models.

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model (%)

Self kernels 19.32 ± 5.21 14.39 ± 3.86

Self & Cross kernels 10.79 ± 4.48 5.88 ± 2.51

DIMOKA et al.862

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