KA ) and a hyperpolarization activated cationic current (I h ). Coupling between soma and dendrites is ensured by conductance g c .

Analysis of intracerebral EEG recordings of epileptic spikes: insights from a neural network model

Sophie Demont-Guignard, Pascal Benquet, Urs Gerber, Fabrice Wendling

Abstract-The pathophysiological interpretation of EEG signals recorded with depth electrodes (i.e. local field potentials, LFPs) during interictal (between seizures) or ictal (during seizures) periods is fundamental in the pre-surgical evaluation of patients with drug-resistant epilepsy. Our objective was to explain specific shape features of interictal spikes in the hippocampus (observed in LFPs) in terms of cell and networkrelated parameters of neuronal circuits that generate these events. We developed a neural network model based on "minimal" but biologically-relevant neuron models interconnected through GABAergic and glutamatergic synapses that reproduces the main physiological features of the CA1 subfield. Simulated LFPs were obtained by solving the forward problem (dipole theory) from networks including a large number (~3000) of cells. Insertion of appropriate parameters allowed the model to simulate events that closely resemble actual epileptic spikes. Moreover, the shape of the early fast component ('spike') and the late slow component ('negative wave') was linked to the relative contribution of glutamatergic and GABAergic synaptic currents in pyramidal cells. In addition, the model provides insights about the sensitivity of electrode localization with respect to recorded tissue volume and about the relationship between the LFP and the intracellular activity of principal cells and interneurons represented in the network. Index Terms-CA1; computational modeling; hippocampus; local field potentials; population spikes I. INTRODUCTION Epilepsy, one of the most common neurological disorders afflicting approximately 1% of the population, is characterized by recurrent seizures that profoundly affect the patients' lives. Over the past decades, clinical and experimental studies have improved our understanding of several underlying mechanisms involved in epilepsies. However, many issues remain unresolved. In particular, difficulties remain in the interpretation of intracerebral EEG signals (depth-electrodes, local field recordings) recorded in patients with drug-resistant epilepsy during the pre-surgical assessment. In most partial epilepsies, two types of "markers" are usually observed in these signals. On the one hand, seizures are characterized by fast oscillations in the gamma [START_REF] Watanabe | Dendritic K+ channels contribute to spike-timing dependent long-term potentiation in hippocampal pyramidal neurons[END_REF][START_REF] Poirazi | Arithmetic of subthreshold synaptic summation in a model CA1 pyramidal cell[END_REF][START_REF] Heilman | Computational models of epileptiform activity in single neurons[END_REF][START_REF] Gasparini | On the initiation and propagation of dendritic spikes in CA1 pyramidal neurons[END_REF][START_REF] Hajos | Modulation of septo-hippocampal Theta activity by GABAA receptors: an experimental and computational approach[END_REF][START_REF] Kali | Distinct properties of two major excitatory inputs to hippocampal pyramidal cells: a computational study[END_REF][START_REF] Li | Computational simulation of the input-output relationship in hippocampal pyramidal cells[END_REF][START_REF] Orban | Intrinsic and synaptic mechanisms determining the timing of neuron population activity during hippocampal theta oscillation[END_REF][START_REF] Hoffman | K+ channel regulation of signal propagation in dendrites of hippocampal pyramidal neurons[END_REF][START_REF] Bernard | Acquired dendritic channelopathy in temporal lobe epilepsy[END_REF][START_REF] Mayer | A voltage-clamp analysis of inward (anomalous) rectification in mouse spinal sensory ganglion neurones[END_REF][START_REF] Lancaster | Calcium-dependent current generating the afterhyperpolarization of hippocampal neurons[END_REF][START_REF] Lancaster | Properties of two calcium-activated hyperpolarizations in rat hippocampal neurones[END_REF][START_REF] Freund | Interneurons of the hippocampus[END_REF][START_REF] Klausberger | Neuronal diversity and temporal dynamics: the unity of hippocampal circuit operations[END_REF][START_REF] Gahwiler | Organotypic slice cultures: a technique has come of age[END_REF] band of the EEG [START_REF] Bartolomei | Pre-ictal synchronicity in limbic networks of mesial temporal lobe epilepsy[END_REF][START_REF] Fisher | High-frequency EEG activity at the start of seizures[END_REF][START_REF] Traub | Fast Oscillations and Epilepsy[END_REF][START_REF] Wendling | Epileptic fast intracerebral EEG activity: evidence for spatial decorrelation at seizure onset[END_REF]. This "fast tonic discharge" gradually changes into slower "clonic activity", mostly in the theta band. On the other hand, between seizures, signals are characterized by the occurrence of so-called "interictal paroxysmal events" that mostly consist of transient spikes or spike-waves. S. Demont-Guignard is supported by the Region Bretagne. The authors also thank Pr Fabrice Bartolomei for human interictal depth-EEG recordings and for his comments and suggestions on this manuscript.

S. Demont-Guignard and F. Wendling are with INSERM, U642, Rennes, F-35000, France and Université de Rennes 1, LTSI, F-35000, France, e-mail: fabrice.wendling@univ-rennes1.fr. P. Benquet is with CNRS UMR6026, Université de Rennes 1, Rennes, F-35000, France and Université de Rennes 1, LTSI, F-35000, France. U. Gerber is with Brain Research Institute; Zurich; CH-8057, Switzerland.

In this paper, we focus on this second type of activity. It has been hypothesised in several studies [START_REF] Alarcon | Origin and propagation of interictal discharges in the acute electrocorticogram. Implications for pathophysiology and surgical treatment of temporal lobe epilepsy[END_REF] that morphological features (amplitude, duration, sharpness) of interictal events may reflect excessive excitability of neuronal networks that generate these events. Therefore, interictal events convey important information regarding the "epileptogenic" properties of neuronal tissue. The objective of the present study is to "decode" this information using a computational modeling approach.

We present a biologically-inspired model of a neural network reproducing the main physiological features (principal cells, interneurons, connectivity) of the CA1 subfield of the hippocampus, a mesial structure of the temporal lobe often involved in temporal lobe epilepsy (TLE). Using dipole theory, this model incorporates a sufficiently large number of cells (in the order of 3000) to allow the simulation of a local field potential (LFP) as recorded with a depth electrode positioned in the anterior hippocampus of an epileptic patient. The model was therefore used to investigate how cell-related and network-related parameters influence the LFP. More specifically, our goal was to characterize the parameters that alter the morphological features of simulated waveforms in response to a stimulation that mimics an afferent volley of action potentials from the CA3 area via the Schaffer collaterals to the CA1 network. Parameter sensitivity analysis was used to derive input values that permitted the model to simulate events closely resembling actual epileptic spikes. In particular, the shape of the early fast component and the late slow component of the epileptic spike could be linked to the relative contribution of glutamatergic and GABAergic synaptic currents in pyramidal cells, controlled, in the model, by modifying the conductances associated with these currents.

In addition, the model provided insights into i) the sensitivity of electrode localization with respect to the recorded volume, ii) the relationship between the LFP and the intracellular activity in principal cells and interneurons and iii) the shape of the network response as reflected in the LFP with respect to the stimulation parameters. Parts of this work have been published previously in preliminary form [START_REF] Demont-Guignard | Interpretation of intracerebral-EEG epileptic spikes from detailed modeling of neural networks[END_REF].

II. A DETAILED MODEL OF THE CA1 SUBFIELD OF THE

HIPPOCAMPUS

We have developed a physiologically and biophysically plausible neural network model of the CA1 subfield of the hippocampus, a sub-cortical structure often involved in temporal lobe epilepsy. It consists of biologically-inspired neurons (pyramidal cells) and interneurons (basket and orienslacunosum moleculare -OLM -cells) interconnected according to available data on the actual circuitry of the hippocampus [START_REF] Andersen | The Hippocampus Book[END_REF]. In line with the many studies reviewed below in section II.A, we used a compartmental modeling approach as well as the Hodgkin-Huxley formalism to describe the membrane properties of neurons. The novelty of our proposed model of a CA1 pyramidal cell (presented in section II.B) lies in the fact that even though it is reduced to two compartments (in line with Pinsky and Rinzel's work [START_REF] Pinsky | Intrinsic and network rhythmogenesis in a reduced Traub model for CA3 neurons[END_REF] on CA3 pyramidal neurons), it accounts for a variety of somatic and dendritic currents of potential interest in epilepsy. Several of these currents are not represented in models published elsewhere. In addition, we validated this cell model experimentally using signals recorded intracellularly from organotypic hippocampal slices (see section IV.A) before studying the effects of network-parameters that lead to the generation of epileptic spikes (section V.B).

A. Background

During the patt decades and since the first biomathematical description of the action potential in the squid giant axon [START_REF] Hodgkin | The components of membrane conductance in the giant axon of Loligo[END_REF][START_REF] Hodgkin | Measurement of currentvoltage relations in the membrane of the giant axon of Loligo[END_REF], computational modeling of single neurons and of networks of neurons has advanced considerably. Classically, in single cell models, compartments are used to represent parameters related to neuronal morphology, to passive (membrane capacitance, axial resistivity, leak conductance, membrane time constant) and to active electrical properties (voltage-gated and ligand-gated ion channels) that are specific to each cell type, and that obey current balance equations (see ( 1)-( 2)-( 3)). Table 1 provides the main features of previously published detailed models that reproduce the activity of pyramidal cells in the CA1 and CA3 subfields of the hippocampus. The first detailed models were developed by Traub and colleagues [START_REF] Traub | Cellular mechanism of neuronal synchronization in epilepsy[END_REF] in the 1980s, and were used to study the impact of network synchronization on the generation of epileptiform activity. Regarding these pioneering studies, it is noteworthy that Pinsky and Rinzel [START_REF] Pinsky | Intrinsic and network rhythmogenesis in a reduced Traub model for CA3 neurons[END_REF] showed that a spatially-reduced version (2 compartments, soma + dendrites) of Traub's CA3 model [START_REF] Traub | A model of a CA3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances[END_REF] (19 compartments: soma + axon + dendrites) can still preserve most of the essential features of the initial model. As depicted in table 1, models differ with respect to structural (number of compartments, presence/absence of the axon) and functional characteristics (transmembrane currents). Indeed, the number of compartments ranges from one [START_REF] Golomb | Contribution of persistent Na+ current and M-type K+ current to somatic bursting in CA1 pyramidal cells: combined experimental and modeling study[END_REF] to several hundreds [START_REF] Migliore | Signal propagation in oblique dendrites of CA1 pyramidal cells[END_REF]. Although the minimal configuration of one compartment for the soma alone can be used, most of the models include at least one compartment for the dendrites. The representation of the axon space is generally not included. It is noteworthy that some of the models presented in table 1 correspond to more advanced versions based on previously presented models. It can also be observed that for both subfields, some currents are always included (voltage-dependent sodium currents I Na , potassium delayed-rectifier currents I KDR ) such as those directly involved in action potential (AP) generation. In some cases, additional currents such as muscarinic potassium current I m , after-hyperpolarisation calcium-dependent potassium currents I AHP are added to represent accommodation of spike frequency.

B. CA1 pyramidal cell: proposed reduced model

We followed Pinsky and Rinzel's approach [START_REF] Pinsky | Intrinsic and network rhythmogenesis in a reduced Traub model for CA3 neurons[END_REF] and developed a 'reduced' two-compartmental model of a CA1 pyramidal neuron illustrated in figure 1. Our rationale was to achieve a compromise between physiological relevance and computing time. In this regard, our model does not provide a very accurate 3D representation of the dendritic tree (which requires a high number of compartments) but allows the simulation of LFPs from networks that include a sufficiently large number of cells. Based on the Hodgkin and Huxley formalism [START_REF] Hodgkin | The components of membrane conductance in the giant axon of Loligo[END_REF][START_REF] Hodgkin | Measurement of currentvoltage relations in the membrane of the giant axon of Loligo[END_REF], equations of conservation of electric charge for the two compartments can be written as follows:

( ) ( )

m m i o n ion dt
According to this formalism, the cell membrane acts as a capacitor Cm connecting the membrane potential Vm (stated Vs for the soma and Vd for the dendrites) to currents passing through the membrane; with I(t) (Is(t) for the soma or Id(t) for the dendrites) the injected input current. For the sum of the

dV I t C I t = +∑ (1)
io n nic curre ts, each current Iion is given by . . .( ) Special attention was paid to the selection of the main ionic channels (voltage-gated and leak) included in the model. In this respect, it differs from previous work in that it accounts for more currents than classical reduced models. The selection of the main ion channels (voltage-gated and leak) was performed based on a review of the literature and was reinforced by data recorded in an experimental preparation (organotypic rat hippocampal slices, see Section IV.A). Current equations were taken from more detailed models published elsewhere and summarized in APPENDIX 1. In brief, the shape of the AP was mainly generated by I Na , I KDR and I Ca (low and high threshold calcium currents). In addition, to replicate physiological firing rates potassium channels activated by intracellular calcium ions were added (I AHP ). Similarly, a rapidly inactivating potassium current I KA , present only in the dendritic compartment [START_REF] Hoffman | K+ channel regulation of signal propagation in dendrites of hippocampal pyramidal neurons[END_REF], was also included in consideration of its potential role in epileptiform activity as well as a hyperpolarization-activated cationic cu within the dendritic compartment [START_REF] Mayer | A voltage-clamp analysis of inward (anomalous) rectification in mouse spinal sensory ganglion neurones[END_REF]]), and I m [START_REF] Lancaster | Calcium-dependent current generating the afterhyperpolarization of hippocampal neurons[END_REF][START_REF] Lancaster | Properties of two calcium-activated hyperpolarizations in rat hippocampal neurones[END_REF].

ion ion m ion I g x y V E = - (2)
( ) ( ) ( ) m x m V β + (4) m x x V V x α α = inf with the time constant ( ) ( ) ( ) m x m x x V V α τ + = 1 (5)

A. CA1 interneuron models: OLM and basket cells

The hippocampal circuitry comprises excitatory principal cells and inhibitory interneurons. Although principal neurons are rather uniform within each hippocampal subfield, a large morphological and functional heterogeneity is typical of interneurons [START_REF] Freund | Interneurons of the hippocampus[END_REF][START_REF] Klausberger | Neuronal diversity and temporal dynamics: the unity of hippocampal circuit operations[END_REF]. We included only the two most common types of GABAergic interneurons in our model of the CA1 network: basket were represented by single compartment models take the literature [START_REF] Hajos | Modulation of septo-hippocampal Theta activity by GABAA receptors: an experimental and computational approach[END_REF].

CA1 network model: connectivity parameters and stimulation

Network simulations included glutamatergic (AMPA and NMDA) and GABAergic synapses. The corresponding equations are provided in APPENDIX 2. Simulated cells were connected to reproduce the CA1 circuit (fig. 2-a andb) in accordance with commonly accepted assumptions (location of synapses and receptors, synaptic strength, connectivity patterns). The percentage of pyramidal cells was set at 80%, that of interneurons at 20%. The number of basket cells was set to equal the number of OLM cells. Regarding connectivity patterns, the number of connections per cell and the maximum distance between two connected cells were fixed according to the literature [START_REF] Andersen | The Hippocampus Book[END_REF]. The probability of a connection was based on a Gaussian distribution: physiologically, the probability of a connection between two neurons is larger when the two cells are close, and is null beyond a maximum distance. The maximum distance depended on the connection type. If d(presynaptic cell, postsynaptic cell) denotes the maximum distance at which the presynaptic cell can connect a given cell, values were adjusted as follows: d(pyr, pyr)=20µm, d(pyr, bas)=500µm, d(pyr, olm)=500µm, d(bas, pyr)=700µm, d(bas, bas)=700µm, d(olm, pyr)=840µm and d(olm, bas)=840µm. selected in CA1, that are contacted by axons firing action potentials and ii) the delay between these action potentials occurring according to a Gaussian distribution. Pyramidal neurons and interneurons were evenly distributed in a 3D space, as shown in figure 2-c. Finally, the whole network was stimulated with an afferent volley of quasisynchronous action potentials coming from an "external array of cells" (fig. 2-d). The objective was to mimic the input from CA3 to CA1. In practice, the stimulation of the CA1 network depends on two parameters: i) the number of cells, randomly and uniformly selected in CA1, that are contacted by ax potentials occurring according to a Gaussian distribution.

III. RECONSTRUCTION OF THE LOCAL FIELD POTENTIAL (LFP)

Here, we assumed that the simulated LFP corresponds to the signal induced by the network at the level of a point extracellular electrode placed in its center. The computation of this simulated LFP was based on dipole theory with the following assumptions. First, due to their spatial arrangement 'in palisades', the pyramidal cells are the main contributors to the LFP. Second, each pyramidal cell is considered as a point source, i.e. a current dipole formed by a sink and a source. i pole associated with a cell i is:

2 4 . i i i i r u d V σ π = (6)
where σ is the conductivity, r i is the distance between cell i and the point of measurement (i.e. the recording electrode), i u is a unit vector on the straight line going from cell i (center of the soma compartment) to the recording point i d , is the dipole moment (figure 2-c).

Under the quasi-static approximation and according to the superposition theorem, the contributions of all pyramidal cells sum up instantaneously at the level of the extracellular recording electrode, which is typically positioned in the middle of the dendritic layer along the somato-dendritic axis c. Therefore, the total potential V d at at the center of the modeled network, as illustrated in figure 2generate the level of the re n cording electrode is given by:

∑ = i V V (7) = i 1
where N is the total umber of pyramidal cells. To compute V, the only variable that must be estimated is the dipole moment i d N [START_REF] Demont-Guignard | Interpretation of intracerebral-EEG epileptic spikes from detailed modeling of neural networks[END_REF]. Regarding the orientation, the dipole was aligned with the main axis of the cell. In our reduced two-compartment model of the pyramidal neuron, we assumed that the sink and the source are located at the level of somatic compartment and the dendritic compartment, respectively. Regarding the magnitude of the dipole moment i d the , we estimated the magnitude of the longitudinal current flowing along the somato-dendritic axis as a function of the de dritic and somatic m ed numerical integration: n embrane potential obtain by

i i c i d i s i S l g V V d . . . - =
where i is the cell index, i s V and i d V , respectively, denote the so atic and dendritic membrane potential, g m approximates a cone of radius R S = 7.5 µm and of height H S = 15 µm (S soma at the ratio S soma /S i = cubator at 36°C. After 2-3 weeks in vi itially increased neuronal spike frequency and cterized by

. It was verified that interictal spikes analyzed in this study were generated at the level of an cornu ammonis). a sampling frequency of 20 kHz. For a network including about 3000 cells, the simulation of a 1.7 second segment of activity required 40 minutes on a quad-core PC computer. c is the coupling conductance between the somatic and dendritic compartments, i l is the distance between the sink and the source, and S i is the total membrane surface. Regarding the numerical values, we chose g c = 1 mS.cm -2 , i l = 157.5 µm (half of the size of the dendritic tree and half of the soma size). For S i , we assumed that the soma = πR(R+√(R²+h²))) and th 0.15 as in [START_REF] Pinsky | Intrinsic and network rhythmogenesis in a reduced Traub model for CA3 neurons[END_REF].

IV. REAL DATA

A. Intracellular data from hippocampal slices

Organotypic slice culture preparation. Hippocampal slice cultures were prepared from 6-d-old Wistar rats as described previously [START_REF] Gahwiler | Organotypic slice cultures: a technique has come of age[END_REF] following a protocol approved by the Veterinary Department of the Canton of Zurich. Briefly, hippocampi were dissected from 6-day-old Wistar rats, and tranverse slices (375 µm thick) were maintained in individual tubes in a roller drum in tro the slice cultures were transferred to a recording chamber on an upright microscope (Axioscope FS, Zeiss, Oberkochen, Germany).

Intracellular recordings. CA1 pyramidal neurons were recorded from with patch pipettes in the whole-cell, currentclamp mode using an Axopatch 200B amplifier (Axon Instruments, Foster City, CA). Slices were superfused continuously at a rate of 1-2 ml/min with normal ACSF at a bath temperature of 32°C. Recording pipettes (3-6 MΩ), were filled with an intracellular solution containing the following: 135 mM K-gluconate, 4 mM KCl, 10 mM HEPES, 10 mM Na2-phosphocreatine, 4 mM Mg-ATP, 0.3 mM Na-GTP, pH 7.2, 291-293 mOsm. Electrophysiological data were acquired and analyzed with pClamp 9.0 software (Axon Instruments). Signals were analog-filtered at 10 kHz and sampled at 100-300 kHz. Liquid junction potentials (10 mV) were deducted offline. Drugs were applied by bath perfusion. 4aminopyridine (4-AP) (50-100 µM) was superfused for 2 to 15 min. 4-AP in subsequently induced epileptiform activity chara the spontaneous generation of paroxysmal depolarization shifts (PDS).

B. Depth-EEG data from human hippocampus

Human interictal data shown in this study were selected from depth-EEG recordings performed in a patient with mesial TLE using intracerebral electrodes (10 to 15 contacts, length: 2 mm, diameter: 0.8 mm, 1.5 mm apart) placed orthogonally according to Talairach's stereotactic method [START_REF] Bancaud | Stereoelectroencephalography, In: Handbook of electroencephalography and clinical neurophysiology[END_REF]. The implantation accuracy was per-operatively controlled by telemetric X-ray imaging. A post-operative CT scan without contrast medium was used to check both the absence of bleeding and the precise 3D location of each electrode contact electrode contact positioned in the hippocampus (anterior

V. RESULTS: MODEL-BASED INTERPRETATION OF EPILEPTIC

SPIKES

As a first step, we evaluated the single pyramidal neuron model by comparing the simulated intracellular activity with real intracellular activity recorded from pyramidal cells in an in vitro model (section V.A). As a second step, we built networks from model cells and studied parameter-related conditions that led these networks to generate transient events (in response to stimulation) resembling actual epileptic spikes (section V.B). Simulations were run with software written in C and objective C. The differential equation systems associated with detailed models were solved with the 4th order Runge-Kutta algorithm (fixed step), with A. Single cell activity: simulated versus real action potentials Once selected, current equations were adjusted and validated in comparison to real recordings (figure 3). The constraint is firstly, to reproduce classic electrophysiological basic characteristics and secondly, to explore the relevance of the change of parameters that leads to transitions from normal to epileptic activity (fig. 3-c, d).

The stimulation protocol consisted in injecting a current into the somatic compartment. The current parameters (intensity, duration, instant of stimulation) were set according to the simulation. The model allows somatic or dendritic activity to be recorded.

For similar stimulation protocols, it can be observed that the single cell model, although reduced to 2 compartments, can generate fairly realistic signals, as exemplified in figure 3-a: the simulated APs reproduces the main characteristics of real APs recorded from CA1 pyramidal cells. It is noteworthy that it is sufficient that the cell behavior mimics the real behavior qualitatively (AP shape and threshold, spike rate adaptation upon depolarizing current and sag plus rebound upon hyperpolarization), since our intent is to analyze the global activity of the CA1 network.

Current-clamp recordings were used to examine two classical manifestations of the contributions of I h on intrinsic excitability: the voltage sag seen during hyperpolarizing current injections and the subsequent rebound excitation [START_REF] Vasilyev | Postnatal development of the hyperpolarization-activated excitatory current Ih in mouse hippocampal pyramidal neurons[END_REF]. Rebound excitation is prominent in CA1 neurons, thus I h contributes to intrinsic excitability to a much greater extent in CA1 pyramidal cells than in, for example, CA3 neurons [START_REF] Vasilyev | Postnatal development of the hyperpolarization-activated excitatory current Ih in mouse hippocampal pyramidal neurons[END_REF]. Therefore, rebound excitability reflects normal behavior of CA1 pyramidal cells. Indeed, when a standard current clamp protocol was used to induce a hyperpolarization from -50 to -100 mV in hippocampal slices, rebound excitation evoked an action potential. When a similar hyperpolarizing protocol is imposed in our computational model, an action potential also appeared during the rebound excitation (fig. 3-b). This spike is not elicited when I h is maintained at 0. This result indicates that, in the computational model, rebound excitation is linked to I h deactivation, as is the case in native cells. This parameter is interesting because I h has been shown to be involved in TLE [START_REF] Dugladze | Impaired hippocampal rhythmogenesis in a mouse model of mesial temporal lobe epilepsy[END_REF].

Parameter sensitivity analysis was performed. Conductance values were selected from the literature (basic values, and channel density along the membrane) and then tested using parameter sensitivity analysis in the reduced model to gauge the impact of each parameter in comparison to the others. An example is shown in figure 3-c, d, where parameters were adjusted to reproduce intracellular experimental data both under control and in 'epileptic' conditions.

We paid special attention to fast inactivating potassium channels (I KA ) because they have been shown to be implicated in TLE [START_REF] Bernard | Acquired dendritic channelopathy in temporal lobe epilepsy[END_REF] and perfusion of the I KA blocker 4AP is used in a classical in vitro model of ictogenesis [START_REF] Perreault | 4-aminopyridine-induced epileptiform activity and a GABA-mediated long-lasting depolarization in the rat hippocampus[END_REF]. A-type potassium current blockade affects back-propagation and increases excitability of cells leading to a switch from normal to bursting activity [START_REF] Magee | Dendritic voltage-gated ion channels regulate the action potential firing mode of hippocampal CA1 pyramidal neurons[END_REF]. As depicted in figure 3-c, d, reducing I KA conductance led the computational model to switch from normal to bursting activity. For the same stimulation protocol, an increase in the number of spikes occurred in the experimental recordings in presence of 4AP or by reducing g KA in the computational model.

B. Network activity: simulated versus real epileptic spikes

To analyze the network response to stimulation and its contribution to the response measured by the recording electrode, a critical question had to be solved: how many pyramidal neurons and interneurons should be included in the network? To address this question, we plotted the "static" component of the LFP as a function of the distance along the X axis (orthogonal to the somato-dendritic Y axis of pyramidal cells), as shown in figure 4-a. This static gain can be easily obtained from equation 6 and 7 by setting the modulus of the dipole moment i d equal to 1, for all pyramidal cells. In this case, the reconstructed local field potential depends only on the location of the cells with respect to the recording electrode and on the conductivity of the volume conductor, as the dependence on membrane potential variations is removed. As expected, the gain is maximal at the electrode location (x = 0) and rapidly decays with distance. As shown in figure 4-a, it was found to drop down to 10% of the maximum at x = -375 µm and x = +375 µm, which corresponds to 50 adjacent cells. We chose this value to estimate the total number of pyramidal cells to include in the simulated network, which consisted of an array including 2500 cells (50 cells along the X axis and 50 cells along the Z axis) with a virtual point electrode positioned at its center. In other words, outside this "volume of sensitivity" around the electrode, we assumed that the contribution of cells to the reconstructed LFP is negligible. Regarding the Y axis, we studied the influence of electrode position on the reconstructed LFP. Results presented in figure 4-b andc show how the amplitude of the network response as well its polarity changes when the electrode is moved along the somatodendritic direction of pyramidal cells. It is noteworthy that a realistic order of magnitude was obtained for the response (in the order of several hundreds of µV to a few mV) when the electrode was placed at the level of the dendritic compartment at a distance of 232.5 µm from the soma.

Finally, as described in section II.D, we assumed that in the CA1 subfield, pyramidal cells represent 80% and interneurons 20% of the total number of cells, and that OLM and basket interneurons represent 50% each of the total number of interneurons, leading to a total number of 3124 interconnected cells: 2500 pyramidal neurons, 312 basket and 312 OLM interneurons, randomly distributed among the pyramidal cells.

Starting from this configuration, we studied network activity (as reflected in the reconstructed LFP) in response to a stimulation consisting of an afferent volley of quasisynchronous APs in axons targeting CA1 pyramidal cells randomly chosen in the network. Stimulation parameters (number of stimulated cells, standard deviation of the delay between incoming APs) were varied (not shown). The results showed that beyond a certain percentage of stimulated pyramidal cells in CA1 (30%) and for increased conductances associated with glutamatergic currents, the simulated LFP could exhibit transient events (fig. 5-b) that closely resemble typical epileptic spikes (fig. 5-a) recorded with depth electrodes in the hippocampus of patients undergoing presurgical evaluation for drug-resistant epilepsy. The examples of real interictal spikes (recorded in one patient) displayed in figure 5-a show that the morphological features (amplitude, duration, more or less pronounced negative wave component following the spike component) can strongly vary in time. To explain these morphological features from the model, we conducted a parameter sensitivity analysis focusing specially on conductance values associated with excitatory (AMPA, NMDA) and inhibitory (GABA) synaptic currents. The methodology was straightforward. For each triplet (gAMPA, gNMDA, gGABA) of conductance values over user-defined intervals, we performed a simulation and measured certain features in the shape of the resulting LFP responses evoked by stimulation, as described in figure 5-c (amplitude of the spike (fast early component) and wave (slower late negative wave) components, total amplitude and duration). We then depicted the measured features as a function of the model parameters using color-coded maps, as shown in figure 6. Two neighboring points corresponded to a modification of either NMDA or GABA conductance values. Variances of the AMPA conductance value are indicated by transitions from one map to another.

Interestingly, for some parameter configurations, we noticed that the model becomes unstable and generates an oscillatory activity that could resemble ictal activity. A detailed analysis of this model behavior (beyond the scope of this study) would be necessary. At this stage, the "ictal-like" activity just appears as black squares in the feature maps. A general inspection of these maps showed that the measured features are parameter-dependant. It also revealed that the "ictal-like" activity was mainly produced, in the model, for increased AMPA and NMDA and decreased GABA conductance values.

Regarding the spike component (A1) alone (fig. 6-a), the results showed that the amplitude was mainly correlated with the AMPAergic synaptic current, as shown in figure 6 (transition from condition 1 or 2 to condition 3). This amplitude increase also appeared when the NMDA conductance value was increased, but to a lesser extent. The effect of changes in GABA conductance was found to be in the opposite direction.

Regarding the negative wave (A2) that follows the spike (fig. 6-b), the model showed that the amplitude was relatively proportional to the amplitude of the spike component and was primarily influenced by AMPA conductance. Moreover, the results also revealed that this negative wave became more pronounced for decreased GABA conductance, as shown in figure 6 (transition from condition 3 to condition 4).

The above observations regarding features A1 and A2 also explain the results regarding the peak-to-peak value (A) of the simulated spikes that decrease for increased GABA conductance and that increase for increased AMPA conductance (fig. 6-c).

Regarding the duration (D) of the simulated spikes (transition from condition 1 to condition 2), the model showed an interesting phenomenon. Indeed, spike duration can strongly increase, but within a specific range of AMPA conductance values. As shown in figure 6-d, for no or a slight increase of the AMPA conductance, the duration of the spike increased with both the NMDA and GABA conductances. However, for higher values of AMPA conductance, the duration of simulated spikes was found to decrease and to return to 'control' values.

Finally, figure 7 illustrates another aspect of the model: the possibility to compare simultaneously extracellular field activity (fig. 7-b) and intracellular activity (fig. 7-a) of pyramidal cells and interneurons. In this example, it can be observed that during the interictal spike, pyramidal cells depolarize and fire several APs (fig. 7-c). This behavior could also be observed in the experimental data (fig. 7-d). At the same time, the firing of interneurons (especially basket cells) increases dramatically. This capability of displaying extracellular/intracellular activity simultaneously is remarkable, in particular because interneurons are difficult to record from in in vivo experimental preparations. Therefore, the model can also be used to bridge "macroscopic" features of LFPs and "microscopic" activities taking place at the cellular level, independent of the cell type represented in the network.

VI. DISCUSSION AND CONCLUSION

Intracellular recordings are reasonably well described in terms of the properties of excitable membranes. In contrast, the interpretation of extracellular signals, in particular those reflecting the activity of neuronal networks, has proven to be a challenging and still unresolved issue. Difficulties mainly stem from the complexity of the recorded networks of neurons. The predominance of nonlinear mechanisms give rise to complex interactions that complicate the relationship between network activity and recorded activity. Computational modeling provides insights into this complexity as biologically-inspired models can act as bridges between sub-cellular/cellular/network parameters and electrophysiological observations [START_REF] Wendling | Computational models of epileptic activity: a bridge between observation and pathophysiological interpretation[END_REF]. As written by G. Buzsáki in the conclusion of his review paper about theta oscillations in the hippocampus [START_REF] Buzsaki | Theta oscillations in the hippocampus[END_REF], "In addition to physiological experiments, several topics could benefit from computational models. For example, modeling extracellular current flow using networks of neurons may yield insights into the relative contribution of spiking and nonspiking neurons to local field potentials".

We believe that the work presented in this paper is in line with this perspective, in the different context of epilepsy. Indeed, this study represents an attempt to identify relationships between "macroscopic" features (amplitude, duration, shape) of interictal events and the basic parameters involved in excitatory and inhibitory synaptic transmission (conductances associated with glutamatergic and GABAergic currents) in neuronal networks.

The main findings of this study are summarized below. First, the model was able to reproduce spikes relatively similar to epileptic spikes, not spontaneously but for a physiologically-relevant stimulation mimicking the input from CA3 to CA1. This result is in line with several experimental studies showing that the spontaneous generation of interictal spikes is abolished when the Schaffer collaterals are cut. This phenomenon is sometimes referred to as the "CA3-driven interictal activity" ( [START_REF] Schwartzkroin | Cellular and field potential properties of epileptogenic hippocampal slices[END_REF], review in [START_REF] Avoli | Network and pharmacological mechanisms leading to epileptiform synchronization in the limbic system in vitro[END_REF]). Aforementioned studies have also shown that CA3 pyramidal neurons can fire quasi-synchronously during interictal activity and that epileptiform patterns originating in the CA3 subfield can propagate to the CA1 area with delays of 5-15 ms. In the model, this effect is reproduced by the volley of quasisimultaneous AP's which reaches a subset of randomly selected pyramidal cells of the CA1 network. We also noticed that the model could generate epileptic spikes for of a sufficiently large subset (i.e. >30% of the total number of cells) of excited CA1 pyramidal cells.

Simulated events showed peak-to-peak amplitudes ranging from several hundreds of µV to 1 mV. These values are compatible with actual amplitude values and confirm that the assumptions we introduced for solving the forward problem (section III) are reasonable.

In this study, we assumed that each pyramidal cell contributing to the recorded LFP behaves as an elemental electrical dipole. We started from the fact that the synaptic activation of a given cell produces either a de-or a hyperpolarization at the soma or the dendrites. The subsequent cascade of modifications at the level of voltage-sensitive channels induces an intracellular current flowing along the longitudinal axis of the cell from the source to the sink formed at the dendrite and at the soma depending on whether the synaptic input is excitatory or inhibitory. This current (which is equal to the current flowing in the extracellular space according to the conservation equation) is computed from the dendritic and somatic membrane potentials at each cell that depend, themselves, on all transmembrane ionic currents we included in both compartments of our pyramidal cell model. In this respect, our study follows studies published elsewhere in which the authors also used the total transmembrane current to calculate the field potential [START_REF] Almeida | Mathematical model of the CA1 region of the rat hippocampus[END_REF][START_REF] Varona | Macroscopic and subcellular factors shaping population spikes[END_REF][START_REF] Erdi | Hippocampal theta rhythms from a computational perspective: code generation, mood regulation and navigation[END_REF][START_REF] Traub | Combined experimental/simulation studies of cellular and network mechanisms of epileptogenesis in vitro and in vivo[END_REF]. In addition, the dipole model allowed us to account for the position and orientation of the cells with respect to the electrode position, as both parameters have also an influence on the recorded field activity. Our study is in line with [START_REF] Murakami | Contributions of principal neocortical neurons to magnetoencephalography and electroencephalography signals[END_REF], in which authors make use of a 19-compartment pyramidal cell model to build a network for CA3 subfield. The net current dipole Qi in each cell (produced by ionic and synaptic currents) is computed as the sum of current dipole Qi,k in each compartment k. Therefore, their approach accounts for longitudinal currents and their dipole solution is equivalent to the one we use in our model that includes 2 compartments at a small distance relative to the distance between cell units and electrode position.

Besides, we mentioned that both the dendritic and somatic membrane potentials of principal neurons contribute to the reconstructed LFP. This implies that the sharp component of the simulated epileptic spike reflects, in part, a contribution of APs generated at the soma of CA1 pyramidal cells in response to stimulation. This finding could be verified as the decrease of voltage-dependent sodium currents at the level of CA1 pyramidal cells led to the reduction of the "spike component" amplitude (not shown).

Second, the model highlights the major role of the AMPA conductance in both the generation of the spike and its amplitude. Our model also shows the influence of the NMDA and GABA conductance on spike amplitude. These results corroborate experimental findings showing that the non-NMDA receptor antagonist CNQX (20 µM) as well as NMDA antagonists abolished interictal spikes [START_REF] Curtis | Cellular mechanisms underlying spontaneous interictal spikes in an acute model of focal cortical epileptogenesis[END_REF]. Regarding spike duration, the model shows that it could strongly increase but within a specific range of AMPA conductance values. Indeed, for a slight increase of AMPA conductance, the wave component was found to be more pronounced (and thus of longer duration) when NMDA and GABA conductances were increased. These results are also in line with experimental data showing that the duration of the late part of the extracellular interictal spike was reduced by the NMDA receptor antagonist APV [START_REF] Curtis | Interictal spikes in focal epileptogenesis[END_REF]. In addition, it was shown in [START_REF] Abegg | Epileptiform activity in rat hippocampus strengthens excitatory synapses[END_REF] that epileptiform activity can be related to the enhancement of AMPA receptor mediated synaptic transmission of CA3-CA1 synapses. This effect is dependent on new insertion of functional AMPA receptors into the postsynaptic sites to strengthen excitatory hippocampal synapses. Moreover, epileptiform activity induced NMDA receptor-dependent long-term potentiation of glutamatergic excitatory synapses. It can be argued that this effect was approximated in a very simple manner by enhancing AMPAergic conductances in our computational model.

Third, interictal spikes have long been considered as a prominent "marker" of epileptogenic networks although the exact relationship between spikes and seizures has been -and is still -a matter of debate [START_REF] Avoli | Do interictal spikes sustain seizures and epileptogenesis?[END_REF][START_REF] Janszky | Spatiotemporal relationship between seizure activity and interictal spikes in temporal lobe epilepsy[END_REF]. These events have been extensively studied during the past decades, especially from scalp EEG recordings where various approaches were proposed for detection (reviewed in [START_REF] Gotman | Automatic detection of seizures and spikes[END_REF]), counting [START_REF] Miller | The meaning of interictal spikes in temporal lobe epilepsy: should we count them?[END_REF] topographical analysis [START_REF] Ebersole | Spike voltage topography identifies two types of frontotemporal epileptic foci[END_REF], localization of sources (reviewed in [START_REF] Michel | EEG source imaging[END_REF]), among other objectives.

Regarding depth-EEG signals, the literature is less abundant as intracerebral recording is only indicated for patients who are being considered for surgery. Methods have also been proposed to detect the origin of spikes [START_REF] Dumpelmann | Visual and automatic investigation of epileptiform spikes in intracranial EEG recordings[END_REF] or to study their propagation, as they often involve several brain structures [START_REF] Bourien | A method to identify reproducible subsets of co-activated structures during interictal spikes. Application to intracerebral EEG in temporal lobe epilepsy[END_REF].

However, to our knowledge, only few studies have accurately analyzed the morphology of human depth-EEG interictal spikes and have attempted to relate these characteristics to possible excitation or inhibition-related processes. Among these studies, the one performed by Alarcon et al. [START_REF] Alarcon | Origin and propagation of interictal discharges in the acute electrocorticogram. Implications for pathophysiology and surgical treatment of temporal lobe epilepsy[END_REF] is of particular interest since the authors found that sharper spikes could be generated close to primary epileptogenic regions. This finding is in agreement with our model since "sharp" spikes are generated for large increases of AMPA conductances, i.e. in hyperexcitable networks that could also occur in epileptogenic areas. Moreover, in the hippocampus, we often observed high-amplitude spikes just before seizure onset. Again, an increase in spike amplitude, in our model, is obtained for large increases of AMPA conductance.

Finally, in this study we restricted our analysis to the effects of specific network-related parameters on the shape of the spike. There are many other parameters to be studied with this model.

For instance, several studies demonstrated a potential role for gap junctions in the generation of epileptic activity, particularly in high frequency oscillations (review in [START_REF] Traub | Gap junctions, fast oscillations and the initiation of seizures[END_REF]. At present, these were omitted in our model since the focus was primarily on interictal spikes. Connectivity patterns could also be considered as free parameters. Indeed, recent studies showed that network excitability is enhanced for particular configurations of connections between cells [START_REF] Morgan | Nonrandom connectivity of the epileptic dentate gyrus predicts a major role for neuronal hubs in seizures[END_REF]. The influence of cell-related parameters should also be studied, as several reports reported that the properties of ion channels can be altered in epilepsy. Indeed, upregulation of excitatory glutamatergic receptors in the dentate gyrus has been observed in both human and animal models of hippocampal sclerosis and may contribute to the development of chronic seizures through increased excitatory transmission [START_REF] Mathern | Human hippocampal AMPA and NMDA mRNA levels in temporal lobe epilepsy patients[END_REF][START_REF] Blumcke | Temporal lobe epilepsy associated up-regulation of metabotropic glutamate receptors: correlated changes in mGluR1 mRNA and protein expression in experimental animals and human patients[END_REF]. Our approach allows these issues to be addressed in future studies as the model represents all of the main ion channels, and the 'reduced' pyramidal cells and interneurons maintain computation times for solving the differential equations at a reasonable level.

APPENDIX

The CA1 pyramidal cell model consists of a somatic compartment with membrane potential s V , and a dendritic compartment with membrane potential . Membrane potentials are governed by the following current balance equations:

d V ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) _ _ s m s s K D Rs s Leak Na CaL AHP SK c AHP BK s m s syn s d s dV C I V I V I V I V I V dt g I V I V I V V I p =- - - - - - - -- -+ s ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) _ _ ( ) 1 d m d d d d d d d Leak Na KDR KA CaT CaR c d d m d hd s y n d s AHP SK AHP BK dV C I V I V I V I V I V I V dt g I V I V I V I V I V V I p =- - - - - - - - - - -- -+ -
where p is the proportion of cell area taken up by the soma. Current equations are provided hereafter. Activation and inactivation kinetics are given by equations 3, 4 and 5 (text, section II.B).

1) Voltage-dependent sodium current [START_REF] Migliore | Role of an A-type K+ conductance in the back-propagation of action potentials in the dendrites of hippocampal pyramidal neurons[END_REF][START_REF] Poirazi | Arithmetic of subthreshold synaptic summation in a model CA1 pyramidal cell[END_REF][START_REF] Bernander | Synaptic background activity influences spatiotemporal integration in single pyramidal cells[END_REF][START_REF] Jung | Prolonged sodium channel inactivation contributes to dendritic action potential attenuation in hippocampal pyramidal neurons[END_REF][START_REF] Hodgkin | A quantitative description of membrane current and its application to conduction and excitation in nerve[END_REF] ( ) 2) Voltage-dependent potassium delayed rectifier current [START_REF] Poirazi | Arithmetic of subthreshold synaptic summation in a model CA1 pyramidal cell[END_REF][START_REF] Bernander | Synaptic background activity influences spatiotemporal integration in single pyramidal cells[END_REF][START_REF] Hodgkin | A quantitative description of membrane current and its application to conduction and excitation in nerve[END_REF] ²( ) 3) Fast inactivating potassium current [START_REF] Poirazi | Arithmetic of subthreshold synaptic summation in a model CA1 pyramidal cell[END_REF] ( )

2 Na Na Na I g m h V E = - ( ) with ( ) ( ) 40
KDR KDR K I g m V E = - . ( ) ( ( ) ) 42
KA K A K I g mh V E = - with ( ) ( ) inf 1 1 m m V α = + and ( ) ( ) ( ) ( ) ( )
max 0.1 (deg ) 1 ,0.1 m m m V qt C V τ β α = + ( ) ( ) ( ) ( ) ( , with ) 3 10
) 1 deg V V Q C m V e ξ α - + = ( ) ( ) ( ) ( ) ( , ) 0.00039 ) 1 deg V V Q C m V e ξ β + = ( ) ( ) ( ) .
( )

V V τ = + ( ( ) ( ) ( 0.003 56 (deg ) V Q C h V e α + = ) ) ( ) h h V V β α = .
4) Hyperpolarization activated cationic current [START_REF] Poirazi | Arithmetic of subthreshold synaptic summation in a model CA1 pyramidal cell[END_REF][START_REF] Magee | Dendritic hyperpolarization-activated currents modify the integrative properties of hippocampal CA1 pyramidal neurons[END_REF] ( )

h h h I g m V E = - with ( ) ( ) ( 90)/8.5 inf 1 1 1 V m e -+ = - + and ( ) ( ) ( ( ) (
) )

145 /17.5 16.8 /16.5 [START_REF] Poirazi | Arithmetic of subthreshold synaptic summation in a model CA1 pyramidal cell[END_REF][START_REF] Magee | Synaptic activation of voltage-gated channels in the dendrites of hippocampal pyramidal neurons[END_REF] In the following, Ca in denotes the intracellular concentration of calcium. We used the following set of standard parameter values. Reversal potentials were E Leak = -70 mV, E Na = 60 mV, E K = -80 mV, E Ca = 140 mV and E h = -10 mV. General parameters were g Leak = 0.18 mS/cm², with 5% noise on I Leak and g c = 1 mS/cm², C m = 1 µF/cm² and p=15%. The conductance values in the control configuration were g Na = 70 mS/cm², g KDR = 6 mS/cm², g CaL = 0.5 mS/cm², g AHP_SK = 0.1 mS/cm², g AHP_BK = 2 mS/cm², g m = 3.1mS/cm² for the soma compartment; g Na = 10 mS/cm², g KDR = 2 mS/cm², g CaT = 1 mS/cm², g CaR = 0.75 mS/cm², g AHP_SK = 0.1 mS/cm², g AHP_BK = 0.1 mS/cm², g m = 0.1 mS/cm², g h = 0.4 mS/cm², g KA = 55 mS/cm² for the dendrites compartment.

1 3 0 2 1 V V m if V mV e e otherwise τ - + + > - ⎧ ⎪ = ⎨ + + ⎪ ⎩ 0 . 5) Voltage-dependent calcium currents
1 5 m m m V V τ α β = + V , ( ) ( ) ( ) ( ) 27

APPENDIX 2

Inhibitory postsynaptic GABAergic current [START_REF] Hajos | Modulation of septo-hippocampal Theta activity by GABAA receptors: an experimental and computational approach[END_REF] . After the pulse (t > t1), C = 0 and .

( )

1 t t
We used the following standard set of parameter values. For pyramidal cells: reversal potentials were E AMPA = 0 mV, E NMDA = 0 mV, E GABA = -75 mV; the conductance values in the control configuration were g AMPA = 8 mS/cm², g NMDA = 0.15 mS/cm², g GABA = 25 mS/cm². For interneurons, reversal potentials were E AMPA = 0 mV, E GABA = -75 mV; the conductance values in the control configuration were g AMPA = 5 mS/cm², g GABA = 1.38 mS/cm².

Figure 1 :

 1 Figure 1: Reduced CA1 pyramidal neuron computational model (two compartments). Each compartment has voltage-dependent sodium currents (I Na ), potassium delayed-rectifier currents (I KDR ), calcium-dependent potassium (I AHP ) currents, a muscarinic potassium current (I m ) and a nonspecific leak current (I Leak ). The soma has in addition an L-type calcium current (I CaL ). Dendrites have T and R-type calcium currents (I CaT , I CaR ), a fast inactivating potassium current (I

  and ii) the delay between these action d β are voltage-dependant functions empirically adjusted to fit experimental data (APPENDIX 1).

Figure 2 :

 2 Figure 2: Network structure and reconstruction of the field potential. a) The network model is intended to reproduce several essential features of the CA1 subfield (hippocampus). b) It includes oriens-lacunosum moleculare (OLM) and basket (BAS) interneurons and pyramidal cells (PYR). Each cell is represented by a compartmental model. The respective percentages of cells and the connectivity patterns (synaptic interactions only) were fixed according to the literature. c) The extracellular field recording is calculated by solving the forward problem from the network to the measurement electrode (V) using dipole theory. d) The CA1 network stimulation was obtained by i) stimulating an "external" array of cells representing CA3 pyramidal cells (direct current injection at the soma) and ii) using evoked APs as the input to the network. The asterisk (*) symbolize randomly selected cells in CA1 that are stimulated. The number of stimulated cells and the delay (Δt) among afferent APs can be varied.

Figure 3 :

 3 Figure 3: Validation of the CA1 pyramidal neuron model. a) Comparison of a real action potential (AP) recorded in a CA1 pyramidal cell (left) obtained from rat organotypic hippocampal slices and a simulated AP generated by the CA1 pyramidal cell model (right). Both plots represent the time course of the somatic membrane potential. b) Membrane voltage responses generated by I h in response to a series of hyperpolarizing current pulses. Traces were obtained using the experimental model (left) and the computational model (right). c) Illustration of the effect of I KA blockade on CA1 pyramidal cell firing rate.Voltage somatic responses in control and in presence of the I KA blocker 4aminopyridine (4AP) (left). The computational model was also stimulated and the inactivating potassium conductance value (g KA ) was reduced from 55 mS/cm² (control) to 15 mS/cm² to mimic the effect of 4AP). d) Average effect on CA1 pyramidal cell firing rate for different depolarizing pulses, in absence (Cont) or presence of 4AP on real neurons (from organotypic hippocampal slices, n=7) (left) or AP simulated (decrease of g KA ).

Figure 4 :

 4 Figure 4: Essential features of the simulated local field potential (LFP). a) "Static" component of the LFP recorded by the point electrode as a function of space. Note the attenuation with respect to the distance along the X and Z axes. b) and c) Relationship between the electrode position (along the Y axis) and the amplitude of the reconstructed LFP. L D is the length of the dendritic compartment; H S is the height of the somatic compartment, and i d dipole

Figure 5 :

 5 Figure 5: Comparison between real and simulated data. a) Typical epileptic spikes recorded with depth-EEG electrodes in hippocampus from a patient with TLE. These spikes were sorted according to increasing amplitude. Spikes are typically characterized by a fast early component followed by a slower negative component (arrows in the lower right box in 5-a). This figure shows the variability of interictal events in terms of amplitude and duration. b) Typical epileptic spike simulated from the model. c) Morphological features measured in simulated spikes and analyzed with respect to model parameters. A: peak-to-peak amplitude, A1: Spike component amplitude, A2: Negative wave amplitude D: Duration.

Figure 6 :

 6 Figure 6: Color-coded maps showing the influence of model parameters (conductances associated with excitatory and inhibitory currents) on morphological features of simulated epileptic spikes (a-d). Features are coded from blue for weak values to red according to large feature values. For some parameter configurations, the model could generate oscillatory activity. In these cases, the results appear as black squares. See results for details. Right: Examples of morphologic changes induced by parameter variations. 1 → 2: increase of gNMDA leads to longer duration of the negative wave. 1, 2 → 3: increase of gAMPA influences the spike amplitude. 3 → 4: decrease of gGABA leads to a more pronounced negative wave following the positive component of the spike.

Figure 7 :

 7 Figure 7: simulated intracellular activity versus extracellular field activity in pyramidal cells and interneurons during an epileptic spike. The early sharp positive component ("spike") of the interictal event is caused by the synchronized activity of pyramidal cells at the very onset of the stimulation.Synchronized APs contribute to the LFP. The early "spike" component is followed by a slower negative wave explained by the sustained activity of interneurons. For this slow component, the synaptic activity of both interneurons and pyramidal cells (mixture of both gabaergic and glutamatergic currents) contribute to the LFP.

7 ) 8 )

 78 Calcium-dependent potassium currents[START_REF] Poirazi | Arithmetic of subthreshold synaptic summation in a model CA1 pyramidal cell[END_REF][START_REF] Destexhe | Synthesis of models for excitable membranes, synaptic transmission and neuromodulation using a common kinetic formalism[END_REF][START_REF] Moczydlowski | Gating kinetics of Ca2+-activated K+ channels from rat muscle incorporated into planar lipid bilayers. Evidence for two voltage-dependent Ca2+ binding reactions[END_REF] Muscarinic potassium current[START_REF] Tiesinga | Computational model of carbachol-induced delta, theta, and gamma oscillations in the hippocampus[END_REF]

TABLE I SUMMARIZED

 I LITERATURE REVIEW OF PYRAMIDAL NEURON MODELS FOR CA1 AND CA3 SUBFIELDS IN THE HIPPOCAMPUS.

			Authors		CA	Number of compartments				Ionic currents
							Dendrites	Soma	Axon			Dendrites	Soma	Axon
	Traub et al.,		3		19					I Na , I KDR , I KC , I KA , I Ca , I KAHP
			1991 [12]
	Pinsky et Rinzel,	3	1	1	0			I Na , I KDR	I Ca , I KAHP , I KC ,
			1994 [8]
	Warman et al.,	1		16					I Na , I Ca , I KDR , I CT , I A , I M , I AHP
			1994 [15]
	Migliore, 1996	1	14	1	1		I f , I Na , I KDR , I h	I f , I Na , I KDR , I h	I Na , I KDR , I KM , I h
				[16]
	Almeida et al.,	1		6					I Na , I K
			1998 [17]
	Kamondi et al.,	1	1	1	0				I Na , I K	I NaP , I KS
			1998 [18]
	Migliore et al,	1		202			I Na , I KDR , I KA	I Na , I KDR , I KA	I Na , I KDR , I KA
			1999 [19]
	Varona et al.,		1		265					I Na , I Ca , I KDR , I M , I A , I KC , I KAHP
			2000 [20]
	Menschik and	3		385					I Na , I KDR , I KC , I KA , I CaT , I CaN , I CaL , I KAHP , I m
	Finkel., 2000
				[21]
	Tiesinga et al.,	3	1	1	0			I Na , I KDR	I Ca , I KC , I KAHP , I M ,
			2001 [22]									I K	O , I NaO
	Sargsyan et al,	1	18	1	0				I Na , I KDR
			2001 [23]
	Poolos et al.,		1		202			I Na , I KDR , I KA, , I h	I Na , I KDR , I KA , I h	I Na , I KDR , I KA , I h
			2002 [24]
	Watanabe et al.,	1		202			I Na , I KDR , I KA	I Na , I KDR , I KA	I Na , I KDR , I KA
			2002 [25]
	Migliore, 2003	1		202			I Na , I KDR , I KA, , I h	I Na , I KDR , I KA , I h	I Na , I KDR , I KA , I h
	Poirazi et al,		1		183		I Na , I KDR , I KA , I h , I CaT , I CaR , I R	CaL ,	I Na , I KDR , I KA , I h ,	I Na , I KDR , I KA , I h ,
			2003 [26]						I AHP(SK) , I AHP(BK) , I m	I CaT , I CaR , I R	CaL ,	I CaT , I CaR , I R	CaL ,
													I AHP(	SK) , I AHP(BK) , I	m	I AHP(SK) , I AHP(BK) ,
													I m
	Heilman and		1	1	1	1	I NaT , I KDR , I KA , , I K	Ca , I CaL , I CaT ,	I	NaT , I KDR , I KA, , I KCa ,	I NaT , I KDR , I CaN
	ochi, 2004 Quattr							I R , I N Ca Ca	I Ca , I T , I R , I CaN L Ca Ca
				[27]
	Gasparini and	1					I Na , I KDR , I KA, , I h	I Na , I KDR , I KA , I h	I Na , I KDR , I KA , I h
	al., 2004 [28]
	Hajòs	et al, 2004 [29]	1	256		0		I Na , I KDR , I K M, I KA , I Ca , I KAHP , I KCT , I h
	M igliore et al. ,	1		202			I , I , I , I h Na KDR KA,	I , I , I , I Na KDR KA	h	I Na , I KDR , I KA , I h
			2005 [14]
	K	ali et Freund	,	1	455		0	I Na , I K	DR R , I KA , I KM , I KC ,	I KAHP ,	I	Na ¸IKDR , I KA , I KM	,
			2005 [30]						I CaT , I CaL , I N , I CaR , I h Ca	I KC , I KAHP , I CaT , I CaN
	G	olomb et al,		1	0	1	0				I Na , I NaP , I KDR , I A, ,
			2006 [13]									I M
	G	olomb et al,		1	0	1	0				I Na , I	NaP , I KDR , I A , I	M ,
			2006 [13]									I Ca , I C , I sAHP
	Li et Ascoli,		1		202			I Na , I KDR , I KA , I h	I Na , I KDR , I KA , I h	I Na , I KDR , I KA , I h
			2006 [31]
	Orbán et al,		1	256		0	I Na , I K , I KA , I KM , I KC , I Ca , I KAHP ,	I Na , I	K , I KA , I KM	, I KC ,
			2006 [32]									I h	I Ca , I KAHP
	The present		1	1	1	0	I , I , I Na KA	AHP(BK) , I AHP(	SK) , I KDR ,	I	Na , I KDR , I AHP(BK) ,
			model							I h, , I	m	, I CaR , I CaT	I	AHP	(SK) , I m	, I CaL

  The three types of calcium currents are

				x	=	0.0853	(	T	+	deg	C	)	2	and
				( ) f z ( ) = ⎨ 1 ⎧ -⎪ ⎪ ⎩ 0.196 ( ) ( ) 2 1 z z z e -( 19.88 if abs z ( ) otherwise 10 < ) ( ( 19.88 /10 4 -. )	) 1
	Type T: with	² g m h CaT ) in ghk V ca CaT I = ( , x = -	0.001 0.001 mM mM ca in + ( ) ( ( 1 . V x in ca e -	(, ghk V ca in ) ) ( ) ) 2 f V x ;

  , it is assumed that a brief pulse of neurotransmitter C occurs in the synaptic cleft (the maximal concentration of C is Cmax between t0 and t1). R represents the fraction of open postsynaptic receptor open. During the pulse (from t = t0 to t = t1), C = Cmax, which gives

													was
	computed as		I	GABA	=	g	GABA	0.1 ( sV E GABA -	) / 3.32	. The activation
	variable = ds dt	10	F	( V	s presyn	was )( ) 1 0.07 s --	s	governed where the transmitter by
	release probability F(V presyn ) was a function of the membrane
	potential ( presyn F V	of ) ( 1/ 1 =	+	e	V -	the / 2 presyn	)	.	presynaptic	neuron
		Excitatory postsynaptic AMPA glutamatergic current [29]
	was ds dt ( presyn 1.1 T = ) = T V	0.1 ( sV E -)( ) AMPA g 1 0.19 s s AMPA --( ) ) 2 /2 2.84 / 1 AMPA I = ( V presyn ( presyn V e --+ .	) / 3.32	with and
		NMDA						current	was	computed	as
	I	NMDA	=	g	NMDA	( R BV	postsyn	-	E	NMDA	)	. Implementation followed
	[73].				B	=	1/(1	+	e	0.062 V -	postsyn	) / ([Mg]/ 3.57)	.	As	the
	( R t t -presynaptic spike arriveswhere ) ( ) inf inf 0 0 R R t R ⎡ = + -⎣ ( ( ) 0 / tau t t R e --⎤ ⎦ ) inf max max 0.072 / 0.072 R C = 0.0066 C + and
	tau R	=	( 1/ 0.072 C	max	+	0.0066	)
				( R t t 1 ) ( ) 0.0066 1 R t e --=	-


Alexandre Bouënard

Marcelo M Wanderley

Sylvie Gibet

ADVANTAGES AND LIMITATIONS OF SIMULATING PERCUSSION GESTURES FOR SOUND SYNTHESIS

The increasing availability of software for creating realtime simulations of musical instrument sounds allows for the design of new visual and sounding media. Nevertheless, from a conceptual and pratical point of view, the question of how these new instruments can be controlled has rarely been addressed in the literature. In this paper, we present a framework for the control of virtual percussion instruments by modeling and simulating virtual percussionists, based on a motion capture database and on a physically-based movement simulation environment. We show the benefits and limits of such an approach as a means of synthesizing new expressive percussion performances.

INTRODUCTION AND MOTIVATION

During the past decades, the design of new digital musical instruments and sound synthesis methods have been widely studied [START_REF] Cook | Real Sound Synthesis For Interactive Applications[END_REF][START_REF] Miranda | New Digital Musical Instruments: Control and Interaction Beyond the Keyboard[END_REF]. Especially regarding percussion-related systems, an important research direction is the development of devices to track performer gestures for controlling sound synthesis processes, such as the Radio Baton [START_REF] Boie | The Radio Drum as a Synthesizer Controller[END_REF], the Korg Wavedrum [START_REF] Rule | Keyboard Reports: Korg Wavedrum[END_REF] or the ETabla controller [START_REF] Kapur | The Electronic Tabla Controller[END_REF]. Despite the availability of various devices, the most accurate hardware for tracking percussive gestures remains camera tracking systems [START_REF] Tindale | A Comparison of Sensor Strategies for Capturing Percussive Gestures[END_REF]. These systems offer an effective method for capturing, analysing and virtually reconstructing the performer's whole body, but they fall short in retrieving the dynamic aspects of playing techniques. Moreover, mapping the recorded motion to sound synthesis processes many times relies on non-intuitive multi-dimensional correspondances [START_REF] Dobrian | The 'E' in NIME: Musical Expression with New Computer Interfaces[END_REF]. Finally, with such methods it is also far from straightforward to go beyond the recorded data and reuse it to synthesize adaptative and realistic new performances.

At the same time, developments in sound synthesis have given rise to various methods to generate percussive sounds. Specifically, physics-based synthesis of percussive sounds has involved the modeling of a hammer [START_REF] Avanzini | Controlling Material Properties in Physical Models and Sounding Objects[END_REF], collisions and sliding excitations [START_REF]Physical Modeling of Impacts: Theory and Experiments on Contact Time and Spectral Centroid[END_REF], and drum skins [START_REF] Chuchacz | Physical Models and Musical Controllers: Designing a Novel Electronic Percussion Instrument[END_REF]. However, their main limitation seems to lie in the way they are controlled. Despite a few early attempts to approach this issue, it is still not clear how to formally relate these models to the excitation by a (real or virtual) performer instrumental gesture, even with the availability of relevant works regarding the study of instrumental percussion gestures [START_REF] Dahl | On the beat: Human Movement and Timing in the Production and Perception of Music[END_REF] and the design of new percussion controllers [START_REF] Aimi | Hybrid Percussion: Extending Physical Instruments using Sampled Acoustics[END_REF]. Only a few previous works have explored the modeling of the equivalent gestural actions thanks to a slowly evolving mechanical model [START_REF] Gibet | Codage, Représentation et Traitement du Geste Instrumental[END_REF], and then the simulation of an articulated arm hitting a vibrating membrane [START_REF] Gibet | Gestural Control of Sound Synthesis[END_REF]. More recent attempts to overcome this limitation include works involving the animation of virtual instrumentalists (or virtual models acting as instrumentalists) [START_REF] Hänninen | Virtual Concert Performance -Synthetic Animated Musicians playing in an Acoustically Simulated Room[END_REF][START_REF] Lytle | Pipe Dream[END_REF], and the synthesis of sounds from rigid body simulations [START_REF] Van Den Doel | FoleyAutomatic: Physically-Based Sound Effects for Interactive Simulation and Animation[END_REF][START_REF] O'brien | Synthesizing Sounds from Rigid Body Simulations[END_REF].

In this paper, we present a framework that combines the use of a motion capture database of real percussion performances with the introduction of a physics-based movement modeling and control method of a virtual percussionist. Our approach differs from the methods for animating virtual characters described above by the presence of a physics layer for animating adaptative and responsive virtual percussionists that control the sound synthesis process. It differs from the contributions aiming at synthesizing sounds from rigid body simulations by focusing on the simulation of instrumental gestures, putting the stress on the mapping between new simulated instrumental gestures and sound synthesis algorithms. This approach allows for the synthesis of new virtual percussion performances.

The paper is organized as follows. Section 2 presents the global architecture of our framework. Details on the simulation of percussion gestures and the mapping to sound synthesis processes are given in section 3. Section 4 presents experiments that have been conducted with our framework, namely the synthesis of percussion gesture units that can be composed for creating new percussion sequences. Finally, section 5 discusses the advantages and limitations of our ap-proach, and concludes with further perspectives.

GLOBAL SYSTEM ARCHITECTURE

The proposed system integrates both visual and sound feedback through the real-time simulation of instrumental percussion gestures from a user-defined sequence of gestures, a "gesture score".

The global architecture involves four steps: a) the planning of a sequence of instrumental gestures using data from a motion capture database, b) the physics simulation of the chosen instrumental gestures, c) mapping strategies between simulated gesture and sound response, and d) the sound synthesis process (Figure 1). The planning step takes advantage of a motion capture database of pre-recorded timpani gestures [START_REF]Enhancing the Visualization of Percussion Gestures by Virtual Character Animation[END_REF]. This database contains the full-body continuous movements of three percussionists performing various exercises following the chosen timpani instrumental gesture typology: different drumstick grips (French or German), beat locations, as well as musical variations (legato, tenuto, accent, vertical accent and staccato).

A refinement to the original motion capture database was implemented by segmenting each multi-beat instrumental performance captured into single-beat units [START_REF] Bouënard | Analysis of Percussion Grip for Physically Based Character Animation[END_REF]. As proposed in [START_REF] Cadoz | Capture, Representation and Composition of the Instrumental Gesture[END_REF], gestural segments are obtained by the physical activity of the interaction instrumentalist-instrument (in our case beat impacts during percussion performances). On the contrary to the generic signal-based framework presented in [START_REF] Cadoz | Capture, Representation and Composition of the Instrumental Gesture[END_REF], we rather represent gestural segments using the timpani intrumental gesture typology described above. It allows for the creation of a "score" at the symbolic gesture level by concatenating several gesture units (Figure 2). Gesture units are then directly derived in signals during the synthesis process.

Gesture Score Segmentation

Editing Assembling Motion Capture Database The simulation step translates the gesture score into motion trajectory inputs that controls the motion of a biomechanical model of a virtual percussionist (section 3). It is divided in two parts: a) a physics-based environment that handles the simulation of motion equations, as well as the modeling of the virtual percussionist, while b) the control loop drives the motion of the virtual percussionist using information provided by the gesture score.

A mapping layer (section 3.3) then makes a correspondence between outputs from the simulation and inputs of sound synthesis processes. The system currently supports a choice of sound generation options using the Open Sound Control (OSC) protocol, from pre-recorded timpani sounds to physical modeling of membranes.

INSTRUMENTAL GESTURE MODELING AND SYNTHESIS

Simulation and Modeling

The motion of a solid of mass m is described and simulated by solving equations ( 1) and ( 2) below at each time step during the simulation. The dynamic forces F M and torques τ M applied on a point M of the solid are processed depending on its current state, including its linear acceleration a M , inertia matrix (or tensor) I M , and its angular velocity Ω.

F M = m.a M ( 1
)
τ M = I M . Ω + Ω.I M .Ω (2)
The physics-based modeling of the virtual percussionist is composed of a set of rigid solids linked by mechanical articulations. The biomechanical properties B of these solids (length, mass, density, inertia) are initialized by motion capture data (Figure 3).

From a score to the physics control of the virtual percussionist

The physics control of the virtual percussionist is achieved by modeling mechanical articulations as damped springs be-tween any two solids, making possible to specify the desired angular state (target angle T ) to be reached between two linked solids. These articulations are parameterized by damping and stiffness coefficients (k s , k d ) that are manually tuned, since there is currently no automatic method for determining them. The physics control loop (Figure 3) involves, for each articulation, its angular state S and angular target T , and processes the torques τ to be applied on the linked solids for reaching the target angle T , as described by equation [START_REF] Avanzini | Controlling Material Properties in Physical Models and Sounding Objects[END_REF].

τ = k s .(S -T ) -k d . Ṡ (3)
These target angles T are directly obtained from the gesture score, which is derived into angular motion trajectories at the articulation level.

The presented physics framework for simulating percussion gestures from a gesture score is based on the physics library Open Dynamics Engine [START_REF] Smith | Open Dynamics Engine[END_REF]. The control algorithms we developped and that are at stake during the simulation are presented in [START_REF] Bouënard | Hybrid Motion Control conbining Inverse Kinematics and Inverse Dynamics Controllers for Simulating Percussion Gestures[END_REF][START_REF]Real-Time Simulation and Interaction of Percussion Gestures with Sound Synthesis[END_REF].

Mapping between Instrumental Gesture Simulation and Sound Synthesis

The physics layer is exploited to express the mapping between the simulation of percussion gestures and sound synthesis methods at the physics level (Figure 4).

From the simulation of percussion gestures, a collision detection module can retrieve the physics features of any contact event. Considering the interaction between simulated percussion gestures and a physics representation of a drum membrane, the collision detection algorithm provides information on the impact position, velocity and force (direction and amplitude). These features are usually the inputs to physics-based sound synthesis algorithms.

RESULTS

We have designed two applications for testing our framework. The first consists of a PureData interface allowing users to choose and explore the simulation of various percussion gesture units and several types of sound feedback. The focus of the second application is the editing and simulation of a score composed of percussion gesture units.

Simulation of Instrumental Percussion Gesture Units and Sound Feedback

A test-bed application has been designed for exploring the simulation of percussion gestures and sound synthesis processes, with the integration of both visual and sound feedback. Users can parameterize the percussion gesture to be simulated from a Pure Data user interface (Figure 5), choosing among five playing variations (legato, tenuto, accent, vertical accent and staccato), two grip modes or varied tempi. As for visual feedback (Figure 9), users can explore the percussion performance space and visualize some features of interest, such as drumstick trajectories and the resulting beat impact locations.

The user interface also offers various types of sound feedback: pre-recorded percussive sounds, signal-based and physics-based sound synthesis. For instance, the physicsbased sound synthesis currently uses a modal synthesis algorithm of a drum membrane from Modalys [START_REF] Adrien | Etude de Structures Complexes Vibrantes, Applications à la Synthèse par Modèles Physiques[END_REF][START_REF] Ellis | Modalys Demonstration[END_REF]. Users can parameterize the properties of the drum membrane (radius size, mass and tension) and tune other parameters related to modal synthesis (particularly the number of modes and the resonance). When using this option of sound synthesis, the percussion gesture simulation provides Modalys with the beat impact position and force according to a oneto-one mapping.

Towards Virtual Instrumental Gesture Composition: Assembling and Simulating Units

In this part, we describe experiments that have been conducted for both physics-based synthesis of new percussion performances and sound synthesis. This includes the study of the simulation variability, as well as the assembling and articulation of percussion gesture units for creating new sequences.

Variability in Simulated Movements

An issue that needs to be taken into account is a possible variability of the resultant movements produced by the virtual percussionist.

Depending on the sequence and speed of the gestures chosen, as well as the fine tuning of the mechanical articulations (c.f. equation 3), simulation artifacts can appear and result in large variations in beat impact position and force. These variations are due to the adaptation of the physics simulation to the constraints in the movement data, and can yield unexpected sound phenomena.

An example is presented in Figure 6, where the virtual musician performs a sequence of four beats played legato with different sets of physical constraints (mechanical articulations). On the top graph, a simulation artifact can be seen on the third beat, that results in a larger sound waveform amplitude and sound intensity, actually masking the last beat. This artifact is removed by changing the physics constraints (damping and stiffness coefficients), as shown on the bottom graph. One can see that the four beats are performed similarly, both in terms of amplitude and intensity 1 .

1 Sounds in both simulations have been obtained using Modalys.

Assembly of Gesture Units and Synthesis of new Percussion Sequences

One of the most interesting outcomes of such framework is the possibility of handling and assembling heterogeneous performances using a combination of a few percussion gesture units. Thanks to the physics simulation of the virtual performer, the issue of gesture articulation between movement units is addressed in part by the physics engine, allowing for a more natural sequence of movements2 .

We have tested the simulation of articulating the same percussion gesture (legato) under an accelerando-decelerando musical variation. As shown in Figure 7, the resulting consistency of the instrumental gesture under such a constraint shows the interest of physics simulation. This approach also has an impact on the required size of the movement database obtained through motion capture by reducing the number of different movement articulations that need to be captured.

The current database allows for a combination of the five gesture unit variations described above, where each can be performed at different positions on the membrane (centre, one-third and rim) and at various tempi.

A second simulation example shows the performance of the score presented on the top of Figure 8. Here, the virtual percussionist is given a task that involves variations in percussion gestures units (legato and accent), as well as in the performance tempo. The bottom part of Figure 8 shows the resulting tip trajectories of the drumsticks during the simulation for each hand. One can notice the variability in the amplitude of gestures (height of the tip of the drumstick) reflecting the different gesture units used, and their durations for the notes performed.

DISCUSSION AND CONCLUSION

Advantages and Limitations

Simulating instrumental percussion gestures for controlling sound synthesis processes presents advantages and limitations. These are of different orders, with respect to the combination of motion capture data and simulation, the mapping between simulation and sound synthesis, and finally the creation of new percussion performances.

Associating a motion capture database and the physicsbased synthesis of instrumental gestures yields an accurate simulation of movements and the retrieval of dynamic aspects of instrumental gestures (beat impacts for percussion performance). It also allows to go beyond pre-recorded motion data by exploring and assembling pre-recorded performance conditions. However, the drawback of such an approach lies in the possible addition of unexpected artifacts to the realism of the captured motion by the physics simulation due to the hard tuning of mechanical joints and/or convergence issues of the numerical algorithms. For instance, increasing the speed of the simulation can introduce errors in the beat impact position or variability in the performance (c.f. Figure 6). Furthermore, our framework depends on the motion clips initially recorded, so that performer tracks are intrinsically part of the resulting simulations. It can be an advantage when considering that performance styles will be preserved, but a limitation when recorded errors are then propagated in the resulting simulations.

Such a physics-based approach considers the mapping between the simulation of percussion gestures and sound synthesis at the physics level. Apart from allowing for a direct mapping between gesture and sound, this choice has the advantage of making clear how physics-based sound synthesis processes can be excited. Moreover, not only physics features can be extracted from the simulation, but also kinematic and energy features. The latter can be of interest for more expressive and complex mappings.

Finally, the synthesis of new percussion performances from the assembly of gesture units yields an intuitive and high level way of controlling a responsive and adaptative virtual percussionist. Nevertheless, assembling instrumental gesture units will not necessarily produce the same level of realism and musical expressiveness as in real performances. This is due to higher-level considerations of instrumental performance that are not taken into account in this model, such as cognitive and expressive strategies, or co-articulation between gesture units. Furthermore, synthesizing a large spectrum of gesture variations necessitates the capture of a large database and the development of adapted motion editing techniques to assembly motion units in a realistic way. Such a physics-based approach however partially eliminates the time-consuming step of building an exhaustive motion capture database, since the simulation itself provides an accurate approximation (in a physics simulation sense) of the gesture co-articulation phenomenon.

Conclusion and Future Work

We presented in this paper a framework for synthesizing virtual percussion performances by using a motion capture database and a physically-enabled environment in which a virtual percussionist can interact with sound synthesis processes. Our approach is shown to be effective for creating adaptive and responsive virtual percussion conditions and to provide physically relevant inputs to sound synthesis models using physical models of drums.

Future work includes the study of the effect of timpani playing variations on the presented parameters. We aim namely at analyzing the effect of beat locations and musical variations on timpani instrumental gestures. We also plan to enrich the current database with examples of gesture dynamics (pp, mf and ff ), as well as with data from different performers.

Figure 1 .

 1 Figure 1. Global system framework.

Figure 2 .

 2 Figure 2. Instrumental gesture planning from the motion capture database: editing a score at the gesture level and assembling gesture units.

Figure 3 .

 3 Figure 3. Instrumental gesture simulation: physics modeling and control of the virtual percussionist which aims to reproduce motion trajectories derived from the score.

Figure 4 .

 4 Figure 4. Mapping between instrumental gesture simulation and soud synthesis.

Figure 5 .

 5 Figure 5. Pure Data user interface: on the left panel users select a drumsctick grip and the playing variation to be simulated, on the middle panel users parameterize the visual feedback, and on the right panel users chose among various types of sound feedback such as sound replay, signal-based or physically-based sound synthesis.

Figure 6 .

 6 Figure 6. Simulation of four beats played legato. A simulation artifact (top) causes much larger sound waveform amplitude and sound intensity on the third beat, compared to a stable simulation (bottom) generating fairly constant amplitude and intensity for all beats.

Figure 7 .

 7 Figure 7. Simulation and articulation between legato beats under an accelerando-decelerando musical variation (height of the tip of the drumstick during the simulation).

Figure 8 .

 8 Figure 8. Score (a) performed by the virtual percussionist, and the resulting height of the right (b) and left (c) tip trajectories of the drumsticks during the simulation.

Figure 9 .

 9 Figure 9. Visual feedback during the simulation: users can explore and visualize the virtual percussion performance space (a), as well as drumstick trajectories (b) and beat impact locations (c).

The resulting articulation is not necessarily equivalent to real performer techniques. It will nevertheless be a physically plausible solution to the problem.

ACKNOWLEDGMENTS

The authors would like to thank F. Marandola (McGill University) and S. Dahl (Aalborg University) for their expert insights on percussion performance, as well as R. Caussé and N. Ellis (IRCAM) for support on Modalys. This work is partially supported by the Natural Sciences and Engineering Research Council of Canada (Discovery and Special Research Opportunity grants) and the International Doctoral College of the Université Européenne de Bretagne.