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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 58, NO. 5, MAY 2011 1303

Nonlinear Dynamic Modeling of Synaptically DrivenSingle Hippocampal Neuron Intracellular Activity

Ude Lu*, Student Member, IEEE, Dong Song, Member, IEEE, and Theodore W. Berger, Fellow, IEEE

Abstract—A high-order nonlinear dynamic model of the input–output properties of single hippocampal CA1 pyramidal neuronswas developed based on synaptically driven intracellular activ-ity. The purpose of this study is to construct a model that: 1)can capture the nonlinear dynamics of both subthreshold activi-ties [postsynaptic potentials (PSPs)] and suprathreshold activities(action potentials) in a single formalism; 2) is sufficiently generalto be applied to any spike-input and spike-output neurons (pointprocess input and point process output neural systems); and 3) iscomputationally efficient. The model consisted of three major com-ponents: 1) feedforward kernels (up to third order) that transformpresynaptic action potentials into PSPs; 2) a constant threshold,above which action potentials are generated; and 3) a feedbackkernel (first order) that describes spike-triggered after-potentials.The model was applied to CA1 pyramidal cells, as they were elec-trically stimulated with broadband Poisson random impulse trainsthrough the Schaffer collaterals. The random impulse trains usedhere have physiological properties similar to spiking patterns ob-served in CA3 hippocampal neurons. PSPs and action potentialswere recorded from the soma of CA1 pyramidal neurons usingwhole-cell patch-clamp recording. We evaluated the model per-formance separately with respect to PSP waveforms and the oc-currence of spikes. The average normalized mean square error ofPSP prediction is 14.4%. The average spike prediction error rateis 18.8%. In summary, although prediction errors still could be re-duced, the model successfully captures the majority of high-ordernonlinear dynamics of the single-neuron intracellular activity. Themodel captures the general biophysical processes with a small set ofopen parameters that are directly constrained by the intracellularrecording, and thus, can be easily applied to any spike-input andspike-output neuron.

Index Terms—Hippocampus, Laguerre expansions, neuronmodel, nonlinear dynamics, Volterra kernels, whole-cell patchclamp.

Manuscript received September 9, 2010; revised October 19, 2010 andOctober 27, 2010; accepted December 28, 2010. Date of publication January13, 2011; date of current version April 20, 2011. This work was supported inpart by the National Science Foundation (NSF) (University of Southern Cali-fornia (USC) Biomimetic MicroElectronics Engineering Research Center), bythe Defense Advanced Research Projects Agency (DARPA) (USC REMINDProgram), by the National Institute of Biomedical Imaging and Bioengineer-ing (NIBIB) (USC Biomedical Simulations Resource), and by the Office ofNaval Research (ONR) under Grant N00014-10-1-0685. Asterisk indicates cor-responding author.

*U. Lu is with the Department of Biomedical Engineering, Center for NeuralEngineering, University of Southern California, Los Angeles, CA 90089 USA(e-mail: [email protected]).

D. Song and T. W. Berger are with the Department of Biomedical Engi-neering, Center for Neural Engineering, University of Southern California, LosAngeles, CA 90089 USA (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TBME.2011.2105870

I. INTRODUCTION

D IFFERENT approaches have been used to mathematicallymodel single neurons in the brain [1]–[3]. The particular

approach used depends largely on the purposes of the modelingstudy [4]–[6]. Some models investigate mechanisms underlyingbiophysical properties of neurons [7]–[13], e.g., compartmentalmodels; some aim to quantify functional interactions among thebroad range of mechanisms found in neurons, i.e., input–outputmodeling [14]–[20]; and others pursue computational efficiencyfor large-scale simulation [21]–[25]. A neuron is an extremelycomplex entity. The goal of the mathematical analysis requiresa choice among modeling methodologies. Achieving one goalwith a specific method often means compromising for others.For example, understanding detailed mechanisms underlyingbiophysical processes is often not compatible with computa-tional efficiency and large-scale simulation. Capturing input–output transformation is usually not congruent with identifyingthe individual role of an electrophysiological mechanism. In ad-dition to modeling purposes, the experimental preparation andthe nature of collected data also contribute to variations of mod-eling methodologies. Our goal, in this paper, is to develop asingle-neuron model that: 1) can capture nonlinear dynamicsof single-neuron subthreshold [postsynaptic potentials (PSPs)]and suprathreshold (action potentials) activities; 2) has a gen-eral structure, and thus, can be applied to any spike-input andspike-output neuron; and 3) is computationally efficient.

The transformation of spike trains between neurons is a com-plex process that involves calcium influx in presynaptic termi-nal, presynaptic vesicle release of neurotransmitter, postsynaptictransduction, synaptic integration, somatic integration and ac-tion potential formation, action potential backpropagation intothe dendritic arbors, and retrograde signaling toward the presy-naptic terminal. All these mechanisms interactively contribute tothe nonlinear and dynamical characteristics of neuron-to-neuronspike train transformation. To characterize this transformation,the concerns are twofold. First, how to design an experimentalparadigm to collect information-rich input–output data sets thatcontain interactions among the mechanisms mentioned as manyas possible? Second, how to construct a model that can char-acterize neuron nonlinear input–output transformation based onthe experimental data with minimum assumptions to avoid thebias of partial knowledge?

To meet these requirements, broadband stimulation trains ofsingle all-or-none pulses were delivered to the synaptic region ofCA1, the stratum radiatum containing Shaffer collaterals (SCs)[see Fig. 1(a)]. The all-or-none stimulation pulses mimic the ac-tion potentials, the most common form of input and output sig-nal in biological neural systems. The interpulse intervals of the

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1304 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 58, NO. 5, MAY 2011

Fig. 1. Experimental paradigm. (a) Schematic diagram of a hippocampal slice.The bipolar stimulation electrode was placed on the CA1 stratum radiatum.Poisson random interval stimulation trains were delivered through a stimulationelectrode. A recording glass electrode was patched on the soma of a CA1pyramidal neuron to record the elicited PSPs and action potentials. (b) Pictureshowing a hippocampal slice with stimulating and recording electrodes. (c)Picture showing a recording electrode patching on the soma of a CA1 pyramidalneuron.

stimulation trains follow a broadband distribution. These broad-band stimuli can elicit broad range of physiological responsesand nonlinearities that are resulted from the interactions of thesubcellular neuron processes mentioned earlier [26]–[29]. Thisparadigm mimics the natural condition in biological nervoussystem and elicits most of the neuronal processes mentionedearlier as opposed to other simpler paradigms, e.g., somatic cur-rent injection stimulation paradigm, which involves a smallersubset of all possible mechanisms.

In this study, whole-cell patch clamp was performed to recordcorresponding intracellular PSPs and action potentials, becauseit provides a high-quality continuous tracing of neuron intracel-lular signal in both subthreshold and suprathreshold responseregimes. The SC-CA1 pyramidal cell system was chosen for thestudy because it is one of the most studied brain region.

In 2007, Song et al. used a similar modeling approach to cap-ture neuron spike train transformations [30]. The data collectedin that study were all-or-none suprathreshold activities (in vivoextracellular action potentials) that limit the biological interpre-tation of the model. One goal of this study is to extend thatapproach to the modeling of intracellular activities that includesboth subthreshold and suprathreshold dynamics and their inter-actions. Similar to the previous model [29], the model structurewas constructed based on three principal neuronal processesthat are common in all spike-input and spike-output neurons(see Fig. 2): 1) transformation from presynaptic spike to PSP;2) action potential generation; and 3) spike-dependent modifi-cation of membrane potential.

The signal flow of the model starts from the entering of presy-naptic spikes to the feedforward blocks (see Fig. 2) that char-

Fig. 2. Model comprises three major components: feedforward Volterra ker-nels (k), a threshold (θ), and a feedback Volterra kernel (H). The feedforwardkernels are up to third order, describing the transformation from presynapticstimulation (x) to PSP (u). The threshold (θ) is a constant membrane potentiallevel, above which output action potentials (yh) are generated. The prethresh-old membrane potential (nonspiking) (w) is the summation of PSPs (u) andspike-triggered after-potentials (a). The operation Σ is defined as superimpo-sition. The model output (y) is the superimposition of prethreshold membranepotentials (w) and templates of action potentials.

acterize the nonlinear transformation from presynaptic spikesto PSPs; PSPs then are passed to a constant threshold, abovewhich action potentials are formed; output action potentialsthen trigger a feedback block that describes the dynamics ofspike-triggered after-potential that, in turn, modifies the mem-brane potential following each action potential. In our approach,both feedforward and feedback blocks were implemented withVolterra models [31], [32]. In a Volterra model, the output sig-nal is expressed in terms of the input signal by means of aVolterra power series. The input–output nonlinear dynamics ofthe system is described by a series of progressively higher orderVolterra kernels that can be directly estimated from the experi-mental input–output data. This data-driven property avoids themodeling errors caused by unknown mechanisms and/or partialknowledge of the system.

II. MATERIALS AND METHODS

A. Experimental Procedures

Rat hippocampal slices were prepared acutely before each ex-periment. Two-week-old male Sprague–Dawley rats were anes-thetized with inhalant anesthetic, Halothane (Halocarbon Lab-oratory, NJ), before standard slicing procedures. Hippocampalslices (400 μm) were prepared by using a vibrotome (Leica VT1000 s, Germany) under iced sucrose solution. A surgical dis-ruption of the connection between CA3 and CA1 was performedon each slice before it was transported to oxygenated bath so-lution for maintenance at 25 ˚C. The sucrose solution contained(in millimoles) Sucrose 206, KCl 2.8, NaH2PO4 1.25, NaHCO326, Glucose 10, MgSO4 2, and Ascorbic Acid 2 at pH 7.5 and290 mOsmol. The bath solution contained (in millimoles) NaCl128, KCl 2.5, NaH2PO4 1.25, NaHCO3 26, Glucose 10, MgSO41, Ascorbic Acid 2, and CaCl2 2 at pH 7.4 and 295 mOsmol.

The experiments were performed on the hippocampal SC-CA1 system. CA1 neurons in the rat hippocampus have twoelaborately branching dendritic trees (basal and apical) thatemerge from the pyramid-shaped soma [see Fig. 1(a)]. Thebasal dendrites occupy the stratum oriens, and the apical den-drites occupy the stratum radiatum (proximal) and the stratum

LU et al.: NONLINEAR DYNAMIC MODELING OF SYNAPTICALLY DRIVEN SINGLE HIPPOCAMPAL NEURON 1305

lacunosum–moleculare (distal). The distance from the stratumpyramidale to the hippocampal fissure (hf) is about 600 μm, andthe distance from the stratum pyramidale to the alveus is about300 μm, yielding a 1 mm distance from end to end. CA1 pyra-midal neurons are covered with about 30 000 dendritic spines.The principal excitatory inputs to CA1 pyramidal cells arrivefrom CA3 through SC. The inputs from CA3 pyramidal neuronsthrough SC form synapses on the apical dendrites in the stratumradiatum and the basal dendrites in the stratum oriens [33], [34].

During whole-cell patch-clamp recording, hippocampalslices were perfused with oxygenated bath solution at 25 ◦C. Abipolar stimulation electrode was placed in the CA1 stratum ra-diatum according to visual cues. Bipolar stimulation electrodeswere made in laboratory with formvar-insulated nichrome wire(A-M Systems, Inc., WA). A recording mircopipette electrodewith 4 MΩ tip resistance patched on the somatic membraneof CA1 pyramidal neuron to record the intracellular PSPs andaction potentials. The recording micropipette electrodes wereproduced by heating and pulling thin-wall single barrel borosil-icate glass tubing (World Precision Instrument, FL) using apipette puller (Sutter Instrument P-80/PC). The internal solutionof the recording electrode contained (in millimoles) Potassium-gluconate 110, HEPES 10, EGTA 1, KCl 20, NaCl 4, Mg-ATP2, and Na3-GTP 0.25 at pH 7.3 and 290 mOsmol.

A programmable stimulator (Multi Channel System, Ger-many) was used to deliver Poisson random interval trains (RITs)with a 2 Hz mean frequency with interspike intervals rang-ing from 10 to 4500 ms [35]–[37]. These stimulation patternscan induce a broad range of physiological mechanisms or pro-cesses that have relatively shorter dynamic ranges, i.e., short-term synaptic plasticity [28], [31]. The 2 Hz mean frequencyis consistent with the in vivo firing characteristics of rat CA3pyramidal neurons [38], [39]. Whole-cell patch-clamp record-ing in the mode of current clamp was performed with the HEKAEPC9/2 amplifier with 10 kHz sampling rate. The intensities ofthe stimulation spike trains were adjusted so that approximately50% of the stimulations induced action potentials. This studyincluded 98 trials of 200 s recordings (400 stimulations in eachrecording trial) in 15 cells from 13 different animals.

B. Modeling Procedures

PSP dynamics, spike generation, and spike-triggered after-potential dynamics are separated in the model structure in aphysiologically plausible manner and are captured by an upto third-order feedforward kernel, a constant threshold, and afirst-order feedback kernel, respectively (see Fig. 2). The feed-forward kernels describe the nonlinear dynamical effects ofsynaptic transmission, dendritic integration, and somatic inte-gration and transform presynaptic spikes to PSPs [9], [40], [41].If a corresponding PSP response is higher than the threshold,a template waveform of an action potential is superimposedto the membrane potential. The formed action potential thentriggers the feedback kernel and generates a spike-triggeredafter-potential that modifies the following membrane poten-tial [1], [11], [15], [42]–[45].

The model (see Fig. 2) can be expressed with the followingequations:

w = u(K,x) + a(H, yh) (1)

y ={

w + action potential, w ≥ θw, w < θ.

(2)

In (1), w represents the prethreshold (nonspiking) membranepotential that is the summation of the output of the feedforwardblock, u, and the output of the feedback block, a. The feedfor-ward block K transforms the presynaptic spike trains x to PSPsu. The feedback block H describes the transformation from theoutput spikes yh to the spike-triggered after-potentials a. In (2),y represents the output of the neuron model, a continuous traceof predicted subthreshold and suprathreshold membrane poten-tials. If w is higher than or equal to the threshold θ, a templateof an action potential is superimposed to w, and the feedbackkernel is triggered by the output action potentials yh; if w islower than θ, y is equal to w.

The output of the feedforward block, u, is expressed with anup to third-order Volterra model as

u(t) = k0 +Mk∑

τ1 =0

k1(τ1)x(t − τ1)

+Mk∑

τ1 =0

Mk∑τ2 =0

k2(τ1 , τ2)x(t − τ1)x(t − τ2)

+Mk∑

τ1 =0

Mk∑τ2 =0

Mk∑τ3 =0

k3(τ1 , τ2 , τ3)

× x(t − τ1)x(t − τ2)x(t − τ3) (3)

where k0 is the zero-order kernel, describing the output of thesystem output when the input is absent, i.e., the resting mem-brane potential. The first-order feedforward kernel k1 describesthe system’s first-order (but not single-pulse) response to x (seethe definitions of response functions in Table I for more expla-nations). The second-order feedforward kernel k2 describes thesecond-order (but not paired-pulse) response to x. The third-order feedforward kernel k3 describes the third-order (but nottriple-pulse) response to x. Mk is the length of the memorywindow.

The spike-triggered after-potential a in (1) is expressed witha first-order Volterra model as

a(t) =Mh∑τ =1

h(τ)yh(t − τ) (4)

where the feedback kernel h describes how an output action po-tential yh triggers an after-potential. The output action potentialtrain yh was defined as

yh ={

1, w ≥ θ0, w<θ.

(5)

C. Laguerre Expansion of Volterra Kernel

The Laguerre expansion method effectively reduces the num-ber of open parameters in the Volterra model by expanding the


TABLE ICONVERSIONS BETWEEN VOLTERRA KERNELS AND RESPONSE FUNCTIONS

Volterra kernels with Laguerre basis functions [32], [46], [47].Both feedforward and feedback kernels are estimated by usingthe Laguerre expansion method as

u(t) = ck0 +L∑

j=1

ck1 (j)vkj (t)

+L∑

j1 =1

j1∑j2 =1

ck2 (j1 , j2)vkj1

(t)vkj2

(t)

+L∑

j1 =1

j1∑j2 =1

j2∑j3 =1

ck3 (j1 , j2 , j3)vkj1

(t)vkj2

(t)vkj3

(t) (6)

where

vkj (t) =

Mk∑τ =0

bj (τ)x(t − τ). (7)

In (6), L denotes the number of Laguerre basis functions;ck0 ,ck1 ,ck2 , and ck3 are the Laguerre expansion coefficients ofthe feedforward kernels k0 , k1 , k2 , and k3 , respectively; andvk

j (t) are the convolutions of the Laguerre basis functions bj

and the input stimulation train x.Similarly

a(t) =L∑

j=1

ch(j)vhj (t) (8)

where

vhj (t) =

Mh∑τ =1

bj (τ)yh(t − τ). (9)

In (8), L denotes the number of Laguerre basis functions andch denotes the Laguerre expansion coefficients of the feedbackkernel h. In (9), vh

j (t) are the convolutions of the Laguerre basisfunctions bj and the output action potentials yh. Laguerre basisfunctions bj is expressed as (10), shown at the bottom of thepage.

where τ is the time epoch value; j is the number order ofbasis functions; and α is the Laguerre parameter (0 < α < 1)determining the rate of exponential asymptotic decline of theLaguerre basis functions.

D. Evaluation and Quantification of ModelPrediction Performance

The model parameters/coefficients are estimated according totwo error measures: 1) normalized mean square error (NMSE)that was used to evaluate the nonspiking PSPs waveform pre-diction and 2) spike prediction error rate (SPER) that was usedto evaluate spike prediction. NMSE is defined as

NMSE =T∑

t=1

(y(t) − y(t))2/

T∑t=1

y(t)2 (11)

where y is the recorded data; y is the predicted data; and T is thetotal number of data points in y and y. In both y and y, actionpotentials are excluded without loss of generality.

SPER is defined as the total number of false predictions di-vided by the total number of stimulations, expressed as

SPER

=number of false-positives + number of false-negatives

total number of stimulations.

(12)

Both false positive and false negative cases are false predic-tions made by the model. False positive refers to the situationthat the model predicts a response event (a PSP response evokedby a stimulation) to form an action potential, but the corre-sponding response event does not form an action potential inrecordings. False negative refers to the situation that the modelpredicts a response event not to form an action potential, but thecorresponding event in recordings forms an action potential.

E. Estimations of Model Parameters

In this model, α, L, ck , ch , and θ are the open parametersthat need to be estimated according to either one of the twoerror measures. The optimization processes are discussed in thefollowing.

1) Laguerre Parameter α: Laguerre parameter, α (0 < α <1), determines the rate of the exponential asymptotic decline ofthe Laguerre basis functions. For each in-sample training trial,

bj (τ) =

⎧⎪⎪⎨⎪⎪⎩

(−1)τ α(j−τ )/2(1 − α)1/2 ×τ∑

k=0(−1)k

(τk

)(jk

)ατ−k (1 − α)k , (0 ≤ τ < j)

(−1)jα(τ−j )/2(1 − α)1/2 ×j∑

k=0(−1)k

(τk

)(jk

)αj−k (1 − α)k , (j ≤ τ ≤ M)

(10)


Fig. 3. (a1) Contour plot of NMSE with respect to the Laguerre parameters ofthe feedforward kernels and the feedback kernel (αk and αh ). (a2) and (a3) Twodash lines intersecting at the global minimum of NMSE are plotted and arrowsindicate the global minimum of NMSE. In this example, the NMSE convergedto the global minimum value 12.648% at αk = 0.972 and αh = 0.910. (b)Choosing the optimal number of Laguerre basis functions (L). In the in-sampletraining trial (dashed line), the NMSE decreased monotonically with increasingnumbers of basis functions. In the out-of-sample prediction trials (solid line),the minimum NMSE was found at L = 3.

we scanned the α values from 0.5 to 1 with a stepping size of0.01, and then, applied the quasi-Newton method [48] for finetuning. The optimal values for feedforward kernels (αk ) andfeedback kernel (αh ) produce a global minimum of NMSE infitting the recorded data [see Fig. 3(a)].

2) Number of Laguerre Basis Functions L: The relation be-tween L and NMSE is evaluated in Fig. 3(b). In the case ofin-sample trainings, the NMSE decreases monotonically as thenumber of basis functions increases. However, in the out-of-sample predictions, the NMSE reaches the minimum at L =3 and rises slightly afterward. This phenomenon is known asthe multivariable overfitting effect. In other words, when thenumber of basis functions is higher than three, it produces over-fitted results that are specific to the training dataset and loses

Fig. 4. Evaluations of model prediction performance with NMSE and SPER.(a) Scatter plot of recorded data and predicted data. In this example, NMSEsof K1 , K2 , and K3 models were 18.3%, 15.4%, and 14.1%, respectively (onlyK3 result is plotted). (b) Estimating the optimal threshold with an ROC curve.In this example, SPERS of K1 , K2 , and K3 models were 28.2%, 23.6%, and20.1%, respectively.

the model generality [49]. The analysis indicates that three isthe optimal number for basis functions.

3) Laguerre Expansion Coefficients ck and ch : The optimalLaguerre expansion coefficients ck , and ch are estimated by us-ing the least-squares method as the product of the pseudoinversematrix of V and the recorded data y (after removing artifacts andaction potentials)⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ck0

ck (1)...

ck (p)ch(1)

...ch(q)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1+p+q)×1

=[(V TV )−1V T]

(1+p+q)×T

⎡⎢⎢⎣

y(1)y(2)

...y(T )

⎤⎥⎥⎦

T ×1

(13)where T is the data length of the recorded data, 1+p is the totalnumber of feedforward convolutions

[V k

j

]T ×1

and their prod-

ucts, q is the total number of feedback convolutions[V h

j

]T ×1

,

and V is the concatenated matrix of all V kj , V h

j , and their appli-cable products, which is expressed as

V = [1, V kj |1≤j≤L , V k

j1V k

j2 |1≤j1 ≤L,1≤j2 ≤j1,

V kj1

V kj2

V kj3 |1≤j1 ≤L,1≤j2 ≤j1 ,1≤j3 ≤j2

, V hj |1≤j≤L ]. (14)

In (13), V T refers to the transpose matrix of V. See Fig. 4(a)for a scatter plot comparison of recorded PSP and predicted PSP.

4) Threshold θ: The thresholds are estimated by scanningthrough the range 0—20 mV (step size 0.01 mV) to find athreshold value that gives the least SPER. Receiver operatingcharacteristic (ROC) curves [see Fig. 4(b)] are plotted with re-spect to the false positive rate (x-axis) and true positive rate(y-axis). The optimal threshold has the closest L1-distance tothe upper left corner (0, 1) in the ROC curve.

F. Reconstruction of Volterra Kernels and Response Functions

The feedforward and feedback Volterra kernels are recon-structed by using the optimal Laguerre expansion coefficients


Fig. 5. Representative clip of recorded data (the upper panel) and the corre-sponding predicted data produced by the K3 model (the lower panel) with anoptimal threshold (dashed line).

ck and ch and the Laguerre basis functions bj as follows [30]:

k1(τ) =L∑

j1 =1

ck1 (j1)bj1 (τ) (15)

k2(τ1 , τ2) =L∑

j1 =1

j1∑j2 =1

ck2 (j1 , j2)2

× (bj1 (τ1)bj2 (τ2) + bj2 (τ1)bj1 (τ2)) (16)

k3(τ1 , τ2 , τ3) =L∑

j1 =1

j1∑j2 =1

j2∑j3 =1

ck3 (j1 , j2 , j3)6

×

⎛⎜⎜⎜⎜⎜⎝

bj1 (τ1)bj2 (τ2)bj3 (τ3)+bj1 (τ1)bj3 (τ2)bj2 (τ3)+bj2 (τ1)bj1 (τ2)bj3 (τ3)+bj2 (τ1)bj3 (τ2)bj1 (τ3)+bj3 (τ1)bj1 (τ2)bj2 (τ3)+bj3 (τ1)bj2 (τ2)bj1 (τ3)

⎞⎟⎟⎟⎟⎟⎠

(17)

h(τ) =L∑

j=1

ch(j)bj (τ). (18)

To examine the effect of a given number of input pulses, weutilized the notion of response functions [31], [49]. As shownin Table I, response functions (r) can easily be calculated fromVolterra kernels (k). The first-order response function r1 de-scribes the single-pulse response elicited by a single input pulse;the second-order response function r2 describes the paired-pulseeffect caused by pairs of input pulses; and the third-order re-sponse function r3 describes the triple-pulse effect caused bytriplets of input pulses. The feedback Volterra kernel is only offirst order; thus, the feedback response function is equal to thefeedback Volterra kernel (h), as in (18). Examples of responsefunctions are plotted in Figs. 6–8.

Fig. 6. Response functions of a K1 model. (a) Single-pulse feedforward re-sponse function (r1 ) is the single-pulse response to an input event. (b) Feedbackresponse function (h) describes the spike-triggered after-potential. The insetsillustrate the relations between input pulses, output spike, and the time epoch τ .

Fig. 7. Response functions of a K2 model. (a) Single-pulse feedforward re-sponse function (r1 ). (b) Paired-pulse feedforward response function (r2 ). (c)Feedback response function (h). The insets illustrate the relations between inputpulses, output spike, and the time epoch τ .

G. Out-of-Sample Prediction

The out-of-sample prediction is performed in two steps. First,the PSP u is predicted as follows:⎡⎢⎢⎣

u(1)u(2)

...u(T )

⎤⎥⎥⎦

T ×1

= [ 1 V k1 · · · V k

p ]T ×(1+p)

⎡⎢⎢⎣

ck0

ck (1)...

ck (p)

⎤⎥⎥⎦

(1+p)×1

.

(19)Second, each PSP response is checked consecutively in the

time order to see if it surpasses the threshold θ. If it does so, anaction potential template is superimposed at the time point tothe surpassing membrane potential, and a spike-triggered after-potential a is added to the subsequent prethreshold membranepotential w, forming a recurrent loop in the prediction process(see Fig. 2). All predictions presented in this paper are out-of-sample predictions, i.e., using one dataset in training and anotherindependent dataset in prediction. The prediction processes are


Fig. 8. Response functions of a K3 model. (a) Single-pulse feedforward re-sponse function (r1 ). (b) Paired-pulse feedforward response function (r2 ). (c)Triple-pulse feedforward response function (r3 ). (d) Feedback response func-tion (h). The insets illustrate the relations between input pulses, output spike,and the time epoch τ .

computationally efficient and were performed with a PC (AMDPhenom 9750).

III. RESULTS

The single-input and single-output (SISO) model described inthis paper can capture high-order nonlinear dynamical charac-teristics of the Schaffer collateral (SC)-CA1 system. Represen-tative recorded and predicted data (produced by the K3 model)are shown in Fig. 5.

To demonstrate how progressively higher order kernels con-tributed to the prediction accuracy, we built three models foreach dataset: K1 includes an first-order feedforward kernel (k1),K2 includes up to second-order feedforward kernels (k1 andk2), and K3 includes up to third-order feedforward kernels (k1 ,k2 , and k3); all models contain a threshold (θ) and a first-orderfeedback kernel (h).

A. K1 Model

The response functions (r1 and h) of a representative K1model are plotted in Fig. 6. In Fig. 6(a), r1 rises abruptly in the

beginning from −0.6 mV, reaches the peak amplitude 10.3 mVat τ = 24 ms, and decays to lower than 1 mV (less than 10% ofthe peak) at 112 ms. In Fig. 6(b), h has a peak amplitude of 6.3mV at the beginning and decays to lower than 0.6 mV (less than10% of the peak) at 121 ms. The K1 model’s average predictionperformance over all datasets in NMSE is 17.9% and SPER is22.4%.

B. K2 Model

The response functions (r1 , r2 , and h) of a representative K2model are plotted in Fig. 7. In Fig. 7(a), r1 rises to the peak10.3 mV at 20 ms and decays to 1 mV at 118 ms. In Fig. 7(b),on the diagonal, r2 starts from −6.17 mV, rises to a minimum0 mV at (60 ms, 60 ms), reaches the peak 1.79 mV at (90 ms,90 ms), and decays to 0.14 mV at (190 ms, 190 ms). In generaldescription, r2 response function is depressive for short-interval(less than 60 ms) input pairs, facilitative for longer interval(more than 60 ms) input pairs, and relatively ineffective to inputpairs that have an interval longer than 190 ms. In Fig. 7(c), hhas a peak amplitude 6.3 mV at 6 ms and decays to 0.6 mV at33 ms. The K2 model’s average prediction performance over alldatasets in NMSE is 15.1% and SPER is 20.2%.

C. K3 Model

The response functions (r1 , r2 , r3 , and h) of a representa-tive K3 model are plotted in Fig. 8. In Fig. 8(a), r1 rises tothe peak 10.39 mV at 22 ms and decays to 0.96 mV at 113ms. In Fig. 8(b), on the diagonal, r2 starts negatively at −1.28mV, reaches the minimum −5.5 mV at (30 ms, 30 ms), rise to0.3 mV at (70 ms, 70 ms), reaches the peak 2.3 mV at (110ms, 110 ms), and decays to 0.19 mV at (220 ms, 220 ms). InFig. 8(c), r3 starts at peak 12.6 mV at the beginning and decaysto 0.95 mV in 50 ms. In general description, r3 is facilitativeif the previous two pulses have intervals less than 60 ms to thecurrent pulse. The facilitative effect of r3 is weak and relativelyineffective when the previous pulses have intervals longer than60 ms. In Fig. 8(d), h has a peak amplitude 5.5 mV at the begin-ning and decays to 0.54 mV at 50 ms. The K3 model’s averageprediction performance over all datasets in NMSE is 14.4% andSPER is 18.8%.

D. PSP Waveform Prediction

The PSP waveform prediction was evaluated with NMSE.The distribution of out-of-sample NMSEs to all datasets areshown in Fig. 9 (N = 98). The average NMSEs to K1 , K2 ,and K3 models are 17.9%, 15.1%, and 14.4%, respectively [seeFig. 9(a)]. Moreover, the NMSE improvement from K1 to K2 is14.2% and from K1 to K3 is 18.7% [see Fig. 9(b)].

E. Spike Prediction

Spike trains are predicted by using the estimated kernels andthe optimal thresholds, as described in Section II. The distri-bution of SPERs of all datasets is shown in Fig. 10. The aver-age SPERs to K1 , K2 , and K3 are 22.4%, 20.2%, and 18.8%,


Fig. 9. Distributions of NMSEs and NMSE improvements with increasingmodel orders. (a) NMSE histograms of K1 , K2 , and K3 . (b) NMSE improvementhistograms.

respectively [see Fig. 10(a)]. The SPER improvement from K1to K2 is 11.2% and from K1 to K3 is 18.7% [see Fig. 10(b)].

IV. DISCUSSION

The nonlinear dynamical single-neuron model described inthis paper is genuinely data-driven. In other words, all modelparameters are simultaneously constrained by using experimen-tal data (intracellular recording) with rigorously defined errormeasures, i.e., NMSE and SPER. No arbitrary manipulation ofmodel parameters in regard to error terms is involved. Moreimportantly, all data constraining the model parameters are de-rived from a single experimental set for which broadband in-put conditions are imposed on the preparation. As argued else-where [31], [50], these represent ideal conditions for modelinginput–output nonlinear dynamics of a neurobiological system.Finally, the model captures subthreshold and suprathresholdactivities and their interactions in a single mathematical formal-ism.

The model described here is a general one. It is a hybrid, com-bining both mechanistic (parametric) and input–output (non-

Fig. 10. Distributions of SPERs and SPER improvements with increasingmodel orders. (a) SPER histograms of K1 , K2 , and K3 . (b) SPER improvementhistograms.

parametric) components. Principles of neuronal processes com-mon to all spike-input and spike-output neurons, e.g., biologicalsignal generation and flow (see Fig. 2), determine the modelstructure. On the other hand, the specific properties that arevariable from neuron to neuron are captured and quantified withdescriptive model parameters that are directly constrained byintracellular recording data. This hybrid representation of bothparametric and nonparametric model components partitions datavariance with respect to mechanistic sources, and thus, im-poses physiological definitions to the model components andfacilitates the biological interpretations of the parameters. Inaddition, this hybrid structure representation conducts the esti-mation power of each model component to specific dynamicsof the designated neuronal processes, producing more accurateestimations as opposed to capturing the neuron input–outputtransformation as a whole with a single nonparametric modelcomponent. This model is general enough to be applied to anyspike-input and spike-output neuron, and flexible enough tocapture neuron-to-neuron differences.


In general, achieving both high accuracy and high efficiencyin a single-model formalism is difficult. High model accuracyusually is achieved by including a large number of open param-eters, though unfortunately, this often reduces computationalefficiency. High computational efficiency usually is achieved bysimplifying the modeling methodology; however, model accu-racy is inevitably compromised. Our results show that the modeldescribed in this study achieved both high accuracy and highefficiency. The average model accuracy in predicting PSP mea-sured by NMSE is 14.4% and average suprathreshold spikingactivity measured by SPER is 18.8%. Moreover, the modelingprocedures can be performed easily using a PC (AMD Phenom9750), and the total number of open parameters is relativelysmall, e.g., ten open parameters in the case of the first-ordermodel (K1), 16 in the case of the second-order model (K2), and26 in the case of the third-order model (K3).

The model shown here is an extension of an earlier approachdeveloped by our laboratory [30]. In that study, Song et al.demonstrated the feasibility of using a similar model structureto capture the nonlinear dynamics when the input–output signalsare extracellularly recorded unitary activities (spikes). Due to thedifferent modeling goals and data-collecting paradigms, thereare three major differences between the neuron model used bySong et al. and the neuron model described in this study: 1) thenumber of model inputs; 2) the noise term before threshold; and3) the parameter estimation method. First, the model by Songet al. utilized extracellularly recorded ensemble of spike trains,and thus, was a multi-input and multioutput (MIMO) model.Since each CA1 neurons can potentially receive inputs frommultiple CA3 neurons, it is reasonable to include all observedinputs in the model, and then, downselect them with statisticalmethods [49]. By contrast, in the experimental preparation ofthis study, recorded CA1 neurons received artificially deliveredstimulation pulses from one stimulation source. In other words,the synapses of the intracellularly recorded CA1 neurons wereactivated simultaneously. This resulted in an SISO formalism.

Second, the extracellular MIMO model by Song et al. andthe intracellular SISO model developed here dealt with noisedifferently. There are two major sources of stochastic activityin a neuron: 1) unobserved inputs and 2) intrinsic randomnessof underlying mechanisms [51]–[53]. Both sources exist in theexperimental context of the extracellular MIMO model. Thus,it is well-founded for the MIMO model to include a noise termto capture those stochastic activities. By contrast, in this study,the synaptic connections between CA3 and CA1 neurons weresurgically eliminated so that no spontaneous activity occurred.This experimental preparation drastically reduced the stochasticlevel of the recorded CA1 neuron system and resulted in a modelstructure without an explicit noise term.

As to the last difference in parameter estimation method,extracellularly recorded all-or-none action potentials do notprovide direct information about subthreshold membrane po-tential. This lack of subthreshold information along with theintroduction of a noise term led the MIMO model to the utiliza-tion of a maximum-likelihood method for parameter estimation.On the other hand, intracellular whole-cell patch clamp tracescontinuously the membrane potential in both subthreshold and

suprathreshold regimes. This information-rich data along witha low noise level enabled the use of least-squares estimation toestimate the model parameters in this study.

Despite the good performance, there are still several possi-ble ways to improve/extend the model described in this report.First, neuron threshold has been reported to be dynamical asopposed to a constant [54], [55]. Although the average SPERof our model was already low, we are compelled to incorpo-rate a dynamical threshold in future model developments tomore completely capture the neuron input–output dynamics.We also expect spike prediction accuracy to be increased bythis extension. Second, to enhance the physiological plausibil-ity of the model, a necessary development would be to ex-tend the current intracellular SISO model to an MISO modelby delivering independent asynchronous stimulation trains todifferent CA1 input areas through multiple stimulation elec-trodes and modifying the modeling methodology correspond-ingly. Third, the stimulation paradigm and modeling method-ology applied here are developed mainly to study short-termsynaptic plasticity. In hippocampal and other cortical neurons,higher stimulation mean frequency may lead to longer changesof the input–output properties [28], such as long-term synapticplasticity. The current modeling strategy then needs to be modi-fied in order to track such long-term changes [56], [57]. Finally,this model can be combined with other modeling/experimentalmethods to study the underlying mechanisms of neurons. Specif-ically, by building nonlinear dynamical models of neuronsas described here with and without a certain pharmacologi-cal manipulation, e.g., application of an agonist/antagonist ofa certain type of ionic channel, we can quantify the func-tional contribution of a certain mechanism to the overall sys-tem behavior and gain further understandings and insights toit [44].

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Ude Lu (S’05) received the B.S. degree in electri-cal and control engineering and the M.S. degree inbiological science and technology from the NationalChiao Tung University, Hsinchu, Taiwan, in 2000 and2002, respectively. He is currently working towardthe Ph.D. degree at the Department of BiomedicalEngineering, University of Southern California, LosAngeles.

From 2002 to 2004, he was a Research Associatein the Institute of Biomedical Sciences of AcademiaSinica, Taipei, Taiwan. His current research interests

include nonlinear system analysis of single neuron, modeling input–output prop-erties of neurological systems, electrophysiology of hippocampus, and long-term and short-term synaptic plasticity.

Mr. Lu is a member of the Biomedical Engineering Society and Society forNeuroscience.

Dong Song (S’02–M’04) received the B.S. degree inbiophysics from the University of Science and Tech-nology of China, Hefei, China, in 1994, and the Ph.D.degree in biomedical engineering from the Universityof Southern California (USC), Los Angeles, in 2003.

From 2004 to 2006, he was a Postdoctoral Re-search Associate at the Center for Neural Engineer-ing, USC, where he is currently a Research AssistantProfessor in the Department of Biomedical Engineer-ing. His current research interests include nonlinearsystem analysis of nervous system, cortical neural

prosthesis, electrophysiology of hippocampus, long-term and short-term synap-tic plasticity, and the development of modeling methods incorporating bothparametric and nonparametric modeling techniques.

Dr. Song is a member of the Biomedical Engineering Society and Societyfor Neuroscience.

Theodore W. Berger (M’03–SM’04–F’09) re-ceived the Ph.D. degree from Harvard University,Cambridge, MA, in 1976.

From 1977 to 1978, he conducted postdoctoralresearch at the University of California, Irvine. From1978 to 1979, he was an Alfred P. Sloan FoundationFellow at the Salk Institute. He joined the Depart-ments of Neuroscience and Psychiatry, University ofPittsburgh, Pittsburgh, PA, in 1979, being promotedthrough to Full Professor in 1987. He is currentlythe David Packard Professor of engineering, a Pro-

fessor of biomedical engineering and neuroscience, and the Director of theCenter for Neural Engineering, University of Southern California, where hehas been a Professor of biomedical engineering and neurobiology since 1992,and was appointed the David Packard Chair of Engineering in 2003. He hasauthored or coauthored more than 170 journal articles and book chapters. Heis the coeditor of the Toward Replacement Parts for the Brain: ImplantableBiomimetic Electronics as Neural Prostheses (MIT Press, 2005). In 1997, hebecame the Director of the Center for Neural Engineering, an organization thathelps to unite USC faculty with cross-disciplinary interests in neuroscience,engineering, and medicine. His current research interests include the develop-ment of biologically realistic, experimentally based, mathematical models ofhigher brain (hippocampus) function, application of biologically realistic neu-ral network models to real-world signal processing problems, very large-scaleintegration (VLSI)-based implementations of biologically realistic models ofhigher brain function, neuron–silicon interfaces for bidirectional communica-tion between brain and VLSI systems, and next-generation brain-implantable,biomimetic signal processing devices for neural prosthetic replacement and/orenhancement of brain function.

Dr. Berger was the recipient of the James McKeen Cattell Award from theNew York Academy of Sciences, the McKnight Foundation Scholar Award,and twice the NIMH Research Scientist Development Award. He was elected aFellow of the American Association for the Advancement of Science. While atUSC, he has received an NIMH Senior Scientist Award, was given the LockheedSenior Research Award in 1997, was elected a Fellow of the American Institutefor Medical and Biological Engineering in 1998, received a Person of the Year“Impact Award” by the AARP in 2004 for his work on neural prostheses, was aNational Academy of Sciences International Scientist Lecturer in 2003, and anIEEE Distinguished Lecturer during 2004–2005. He received a “Great Minds,Great Ideas” Award from the EE Times in 2005, and in 2006, was awardedUSC’s Associates Award for Creativity in Research and Scholarship.

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