Multisynaptic activity in a pyramidal neuron model and neural code

BioSystems 86 (2006) 18–26

Multisynaptic activity in a pyramidal neuron model and neural code Francesco Ventriglia∗ , Vito Di Maio Istituto di Cibernetica “E. Caianiello” del CNR, Via Campi Flegrei 34, 80078 Pozzuoli (NA), Italy Received 15 December 2005; received in revised form 19 February 2006; accepted 22 February 2006

Abstract The highly irregular firing of mammalian cortical pyramidal neurons is one of the most striking observation of the brain activity. This result affects greatly the discussion on the neural code, i.e. how the brain codes information transmitted along the different cortical stages. In fact it seems to be in favor of one of the two main hypotheses about this issue, named the rate code. But the supporters of the contrasting hypothesis, the temporal code, consider this evidence inconclusive. We discuss here a leaky integrateand-fire model of a hippocampal pyramidal neuron intended to be biologically sound to investigate the genesis of the irregular pyramidal firing and to give useful information about the coding problem. To this aim, the complete set of excitatory and inhibitory synapses impinging on such a neuron has been taken into account. The firing activity of the neuron model has been studied by computer simulation both in basic conditions and allowing brief periods of over-stimulation in specific regions of its synaptic constellation. Our results show neuronal firing conditions similar to those observed in experimental investigations on pyramidal cortical neurons. In particular, the variation coefficient (CV) computed from the inter-spike intervals (ISIs) in our simulations for basic conditions is close to the unity as that computed from experimental data. Our simulation shows also different behaviors in firing sequences for different frequencies of stimulation. © 2006 Elsevier Ireland Ltd. All rights reserved. Keywords: Pyramidal neuron; Irregular firing; Neural code; Computer simulation; Synaptic constellation

1. Introduction The information arriving to brain from external and internal environment must bear the minimum ambiguity possible to be meaningful. By looking at the single neuron, it seems clear that the information is encoded in the spiking activity, but not in single spikes. In fact, spikes present in general the same time-course in a single neuron, while differences in firing sequences can be found for different stimuli. Moreover, different neurons can produce different patterns of firing when stimulated ∗ Corresponding author. Tel.: +39 081 867 5141; fax: +39 081 8042519. E-mail address: [email protected] (F. Ventriglia), [email protected] (V. Di Maio).

by inputs in the appropriate modality. Hence, it holds that the most probable way neurons code information has to be attributed to spike sequences. In peripheral neurons, and mainly in receptors, the relation between spike sequences and stimuli is more evident being the frequency of spikes related to the stimulus intensity by mathematical laws. These neurons, however, show a rather simple behavior since usually they respond to preferred stimuli and have very few inputs. Different is the case of cortical pyramidal neurons, the main population of excitatory neurons of the neocortex, which receive a large number of synaptic inputs, both excitatory and inhibitory, some of them coming from different regions of the neocortex as well as from deep nuclei and have a very complex activity. In this case, how the output is related to the input is hard to clarify, being also the input very difficult to

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observe and describe due to its complexity. One of the most striking result about the activity of cortical pyramidal neurons derives from electrophysiological studies of visual areas in awake brain of macaque monkey. The time sequences of spikes observed in these neurons are so highly irregular to support the idea of a predominant influence of randomness on their genesis (Softky and Koch, 1993; Shadlen and Newsome, 1994, 1995). In fact, a Poisson process (a typical example of stochastic processes) can adequately describe the spike sequences observed in cortical pyramidal neurons. The randomness of the inter-spike intervals (ISIs) seems to imply that information cannot be coded in the temporal pattern of spikes. Hence, it seems that the firing frequency (averaged on appropriate time intervals) can be considered as a good candidate for coding information (Shadlen and Newsome, 1998). This it the base of the rate (or frequency) code view. In such a framework, the neurons are considered as integrate-and-fire devices which integrate all the inputs (excitatory and inhibitory) arriving from dendritic and somatic synapses. The balancing of the effects of all the inputs at the hillock determines the firing of the neuron as well as the ISIs sequence. Vice versa, a more recent hypothesis assumes that the precise spike times in spike trains, or the inter-spike interval patterns, or the times of the first spike (after an event) are the possible bases of the neural code. It is labeled as the temporal code hypothesis and is linked to the view of a neuron as a coincidence detector (Konig et al., 1996). The main motivation for this view is the assumption that the transmission of information is related to the synchronous activity of local populations of neurons and, consequently, the detection of coincidences among the inputs is the most prominent aspect of the neuronal function (Abeles, 1982, 1991; DeWeese et al., 2003; Softky, 1995). Some authors considered also the possibility that brain uses not a single coding scheme but a continuum of coding procedures ranging from rate to temporal (Tsodyks and Markram, 1997). The non-synchronous activity of tens of thousand synapses, placed at different distance from the hillock and producing Post Synaptic Potentials (PSPs) with peak amplitude stochastically distributed, has been considered as an appropriate support on which the neuronal machinery can produce sequences of spikes with stochastically distributed ISIs. A rich investigation field, built on stochastic models of neuronal activity, arose from the early findings (Gerstein and Mandelbrot, 1964; Lansky, 1984; Ricciardi, 1994; Ricciardi and Ventriglia, 1970; Tuckwell, 1975, 1989). Moreover, several attempts by experimental, modeling–computational and mixed experimental–modeling–computational methods,

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have been carried on to identify the causes of the high irregularity of the firing patterns in pyramidal neurons of cerebral cortex. In some experiments on brain slices, synthetic electrical currents, constructed in a way to simulate the true synaptic activity, have been applied to soma of pyramidal neurons in order to obtain ISIs as irregular as those produced by naturally stimulated neurons (Stevens and Zador, 1998). On the other side, several computational models, with different level of complexity, have been proposed for the same purpose (Softky and Koch, 1993; Shadlen and Newsome, 1994, 1995; Salinas and Sejnowski, 2000; Kuhn et al., 2004; Ventriglia and Di Maio, 2005; Zador, 1998). In some models, the variability of synaptic input has been singled out as the cause of the output variability. In others, the main focus has been given to the dynamic properties of the neuron receiving the stimuli. However, both experimental and computational results still give contradictory interpretations. The main criticism to the theoretical–computational investigation of the features of the neuronal firing is that the models used to describe pyramidal neuron activity are too simple with respect to the biological process they want to simulate and fail to capture relevant information. Stochastic models of integrate-and-fire neuron, for example, reduce to a white noise the effects on membrane potential at the hillock of inputs arriving from synapses of distal dendrites (Di Maio et al., 2004 among many others). In the present paper, we made an attempt to formulate a biophysically realistic model of integrate-and-fire neuron where the contributions of all the synapses to the evolution of membrane potential at the hillock are better considered. To this aim, we have build a geometrical structure of a pyramidal neuron by taking into account the position of all excitatory and inhibitory synapses impinging on such a neuron by using data from literature on hippocampal neurons. In addition, we have used parameters for the description of Excitatory and Inhibitory Post Synaptic Currents (EPSCs and IPSCs) at each synapse, which have been derived from electrophysiological (Forti et al., 1997; Liu et al., 1999; McAllister and Stevens, 2000) and computational data (Ventriglia, 2004; Ventriglia and Di Maio, 2003). The results of several computational experiments, carried out to get information on the causes of the high irregularity of the pyramidal neuron firing and to obtain insights into the nature of the neural code, are discussed here. 2. Model To study the properties of firing sequences of pyramidal neurons, we constructed a model of neuron by using anatomical information from pyramidal neurons in CA3

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(and CA1) field of the hippocampus for which a rather complete description of dendritic structure and of synaptic distribution is available. A general description of such a neuron can be made by dividing it in layers according to the anatomical position of the neuronal components in the hippocampal fields. In this way, our model was composed by a set of modules each having its set of inhibitory and excitatory synapses which were arranged according to data from literature. For each simulation, the activation frequency of both excitatory and inhibitory synapses for each module was “adjusted” so that our neuron model had a basic firing frequency similar to that of cortical pyramidal neurons. 2.1. Model of pyramidal neuron Information from literature on the the main structure of hippocampal pyramidal neurons divides dendrites of pyramidal neurons in CA3 field of Hippocampus in four main layers: Oriens, Lucidum—formed by the mossy synapses of axons coming from granular neurons in Dentate Gyrus, Radiatum and Lacunosum-moleculare. Another layer in CA3, the stratum Pyramidal, contains the somata of pyramidal neurons. In the CA1 field, the Lucidum layer is lacking. Different authors have computed the length and spatial distribution of the pyramidal dendrites and have calculated also the number of synapses in each layer according their nature (inhibitory or excitatory) as well as the number and nature of synapses on the soma and the initial segment of the axon (Amaral et al., 1990; Gonzales et al., 2001; Matsuda et al., 2004; Megias et al., 2001). The gross, total numbers for CA1 are reported in: 31,000 excitatory synapses and 1700 inhibitory synapses (Megias et al., 2001). As regards to pyramidal neurons in CA3, the most recent article with quantitative data (Matsuda et al., 2004), reports space densities and layer distribution for the two classes of synapses. These values have been utilized to compute the distribution of synapses on a single neuron by using the result that 88% are excitatory and 12% are inhibitory. The percentages of excitatory synapses in different layers have been computed and so we know that they are almost 30% in Lacunosum, 28% in Radiatum, 18% in the Lucidum, 1% on Soma and 23% in Oriens. Percentage of inhibitory synapses are: 33% in Lacunosum, 19% in Radiatum, 10% in Lucidum, 9% on the Soma and 29% in Oriens. The distribution of the inhibitory synapses discloses that about 89% are positioned on dendritic Shafts, 9% on Soma and 2% directly act on the initial segment of the Axon (i.e., very close to the hillock). By using the total values obtained for the synapses in pyramidal neurons of CA1 as reference values also for pyramidal neurons

of CA3, and dividing it in 88% excitatory and 12% inhibitory, from the above percentage we can compute the following numbers of synapses for a pyramidal neuron of CA3. The excitatory synapses are: 9300 in Lacunosum, 8700 in Radiatum, 5500 in the Lucidum, 300 on the Soma and 7000 in the Oriens. The inhibitory synapses results to be: 1300 in Lacunosum, 750 in Radiatum, 400 in the Lucidum, 350 on the Soma and 1150 in the Oriens. Taking into account these values, the first step of simulation consists in arranging the synapses into the different layers (modules in our model). For this purpose, each module is considered as a geometrical space according to its anatomical shape. Axon (initial segment), Lucidum and Shaft are considered as cylinders of a given length and radius. Synapses belonging to these modules are placed on the surface of the cylinders. Soma is considered as a sphere with two poles, one on the top (where Lucidum has its base) and one on the bottom. From the bottom pole, where we fixed the origin (x0 , y0 , z0 ) of the space coordinate system, Axon originates. The apex of Oriens was made coincident with the origin of the system. The hillock is then considered as the point P : x0 = 0, y0 = 0, z0 = 0. Also the other modules have their 3D shape. Radiatum and Oriens have been considered as cones: the first with the apex fixed on Lucidum and the other with the vertex coincident with the hillock. Lacunosum is considered as a semi-cone starting at a fixed position over Radiatum. Synapses are then placed in each module following its 3D shape. They are arranged according to a uniform distribution along the x, y, z axes, limited by the 3D shape of the module. As shown in Fig. 1, just the simple positioning of synapses in the 3D spaces provides an outline of the structure of the simulated neuron. Once synapses have been positioned, the distance of each of them from the hillock is computed. To account

Fig. 1. Dendritic tree.

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for the different branching of dendrites, we introduced a parameter called tortuosity coefficient, TC. The TC was used as multiplicative factor for the distance of each synapse and it was chosen according to a Gaussian distribution (N(µ, σ) : µ = 1, σ = 0.2 ). Hence, the distances, corrected by this coefficient, are converted in units of λ (the space constant of the dendritic tree) in order to compute the filtered currents at the hillock according to the equations of the following section.

currents at the soma, next equation, can be used to compute the contribution of each inhibitory and excitatory synapse to the membrane voltage at the hillock:

2.2. Mathematical description

Here, L is the distance between the site of the synapse and the the hillock, in units of λ (the space constant of the dendrite), T the time in units of τm (the membrane time constant) and k is an appropriate constant related to the peak amplitude of the AMPA current. By summing excitatory and inhibitory currents at the hillock we obtain the current I(t):

The computation of the Excitatory (and Inhibitory) Post Synaptic Potential (EPSP and IPSP) produced at the axon hillock by the activation of a generic excitatory (or inhibitory) synapse located on a dendrite (or shaft or soma or initial segment of axon) was based on the method described by Kleppe and Robinson (1999). They computed the unknown activation time course of AMPA receptors of an excitatory synapse on a dendrite by analyzing the time course (recorded at the soma) of the co-localized NMDA receptors. They assumed that the opening time of single NMDA ionic channels is so short that the time course could be considered as a step function. Hence, they computed the filter response of the dendrite to an impulse function and to a step function. In such a way, by using the time course of currents recorded at the soma, they obtained the AMPA phase currents at the synapse. Because from data in literature (Forti et al., 1997; Liu et al., 1999) and our previous investigations on the characteristics of single excitatory synapses of brain (Ventriglia, 2004; Ventriglia and Di Maio, 2003), we know the time course of AMPA currents at the synapses, we inverted the procedure and computed the unknown time course of their filtered effects (currents) at the hillock. Since for inhibitory currents at the synapses conveyed by GABAA receptors we have not so clear results as those for AMPA receptors, we assumed that the IPS currents had the same shape (but opposite direction) of the EPS currents. This procedure seems fair also because by electrophysiological recording from soma of hippocampal pyramidal neurons we know that the time-course of the two currents is the same, except for the sign (Miles, 1990). The following equation describes the time course of currents at the synapse:
    −t −t I(t) = K exp − exp (1) τ2 τ1 where τ1 is the activation time constant, τ2 the decay time constant and K is a scaling constant to obtain the appropriate peak amplitude. The time course of filtered

kL I(T ) = √ 2 (π)  T exp [(u − T )/τ1 ]− exp [(u − T )/τ2 ] du. × u3/2 exp [u + (L2 /4u)] 0 (2)

I(t) =

n  i=1

Ii (t) +

m 

Ij (t)

(3)

j=1

where Ii (t) is the the current furnished by the ith excitatory synapse at time t and Ij (t) is the current furnished by jth inhibitory synapse at time t being n and m, respectively, the number of excitatory and inhibitory synapses having effects at the hillock at time t. The Post Synaptic Potential at the hillock is computed by introducing I(t) in the following differential equation: d V (t) + [V (t) − Vr ]G − I(t) = 0. (4) dt Here, V (t) is the membrane potential, C the membrane capacitance, G the membrane conductance and Vr (= −70 mV) is the resting potential, while a value of −88 mV has been used for the maximum hyperpolarization level. Typical values for the other parameters can be found in Segev et al. (1989). Dividing by G, we obtain:

C

d I(t) V (t) = −[V (t) − Vr ] − . (5) dt G This is translated in the following discrete time equation to compute numerically the PSP, V (t):   ∆ ∆ + V (t + ∆) = V (t) 1 − (6) I(t) + Vr∆ τm Gτm

τm

where Vr∆ is the constant

Vr ∆ τm .

3. Simulation of a CA3 pyramidal neuron The simulation procedure consisted in placing all the synapses on the pyramidal neuron model, in activating them and in computing their effects at the hillock. The times of activation of each excitatory (and inhibitory)

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synapse have been computed according to a Poisson distribution, whose mean frequency (chosen from data in the literature for cortical pyramidal neurons) can vary across the computational experiments. The amplitude of current peaks at each synapse, for each activation time, has been chosen depending on a distribution with a positive skewness which considered both experimental data (Forti et al., 1997; Liu et al., 1999) and computational results (Ventriglia, 2004). The synaptic currents, both excitatory and inhibitory, had then a peak amplitude of 25 ± 35 pA (mean ± standard deviation). At any time step (0.01 ms), the contribution of each synapse to the current arriving at the hillock, computed by using Eq. (2), was summed up and the voltage was computed by Eq. (6). When the voltage crossed a threshold value (which for simplicity has been considered constant with a value of −45 mV), the neuron produced a spike. In a first approximation, spikes are not modeled according to Na+ and K+ channel activation and deactivation as in Hodgking and Huxley model. Nevertheless, each spike is not simply a discontinuity point in the membrane voltage time-course, as usually assumed in simplified models of leaky integrate-and-fire neurons, but after a firing the membrane potential went in a slight hyperpolarization (at a value of −75 mV). During the subsequent refractory period, with a duration of 15 ms, the neuron remained unable to react to incoming synaptic currents, being the I(t) in Eq. (6) forced to 0. At the same time, the membrane potential increasing according to Eq. (6) mimicked a repolarizing phase. At the end of the refractory period, the neuron became again able to react to the synaptic activity. In this paper, we present the results of simulations aimed both to test the basic condition activities and to investigate the effects of the increasing of the activation frequency in excitatory synapses of specific modules. As an example of the localized over-stimulation, the case of Radiatum module has been discussed. In fact, we performed two set of simulations in which, from the basic stimulation frequency of 1 Hz, the frequency was raised in this module to 3 or to 4 Hz during a short period of time (by a common abuse of notation we used in the present article Hertz instead of spikes/s for spiking frequency). All other parameters remained constant as well as the structure and the positions of the synapses. For each simulation, we computed the ISI distribution, the mean ISI, the standard deviation and the CV (i.e., the coefficient of variation of the distribution of ISIs, defined as the standard deviation σISI divided by the mean µISI : ). This last parameter is usually considered as CV = µσISI ISI an evaluator of the neuronal firing irregularity. Also a different statistics, the Fano factor, has been utilized to measure the firing variability. It is based on the spike

count N (and not on the ISI) and is defined as the ratio between the variance (VARN ) and the mean (µN ) of N the spike count: FF = VAR µN . It is used to measure the inter-trial variability (Stevens and Zador, 1998). 4. Results The basic activity of the pyramidal neuron model was tested in a series of computer simulations. Several computations have been carried out by modifying some meaningful parameters of the input, such as the mean and the standard deviation of the peak amplitudes for EPSC and IPSC or the frequency of Poissonian inputs to excitatory and inhibitory synapses. A set of 18 simulations, each producing 20 s of neuronal activity, has been carried out under constant conditions. The same synaptic structure, the same input frequency on excitatory (1 Hz) and inhibitory (8 Hz) synapses, the same statistical parameters of the peak amplitudes of synaptic currents have been used, only the seed of the random number generator being changed. The statistical analysis of the different firing series showed that the firing frequencies ranged from 0.35 to 0.95 Hz, with a mean value of 0.70 Hz and a S.D. of 0.15 Hz, while CV had values in the range 0.61–1.18 with mean value of 0.83 and S.D. of 0.47. The Fano factor, computed to measure the inter-trial variability of all the firing series, had a value of FF = 0.64. The cumulative histogram of all the ISIs of the 18 runs is reported in Fig. 2. A sample of one of the firing sequences of this set is presented in the top panel of Fig. 3. In this specific, instance the mean spiking frequency of the simulated neuron was 0.85 Hz and the CV of ISIs was 0.95. The results of a different computational experiment, carried on by changing only the frequency of activation of the

Fig. 2. Cumulative histogram of ISIs for 18 runs of 20 s simulated CA3 pyramidal neuron activity.

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Fig. 3. Twenty seconds of simulated pyramidal neuron activity for two different combination of synaptic input frequencies—top panel: excitatory 1 Hz and inhibitory 8 Hz; bottom panel: excitatory 1.25 Hz and inhibitory 10 Hz.

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hibitory synapses and depends greatly on the respective frequency of activation. Other computational experiments investigated the effects produced on the firing series by variations of the activation frequency (mean of the Poissonian distribution) of the synapses in specific modules. The procedure runs as follows. At an assigned time and for a specified period of time – usually 1 s – the activation frequency on a determined percentage of the excitatory synapses within a selected module was raised from the base value (1 Hz) to higher values (3 or 4 Hz). The effects of the variation of the percentage of excitatory synapses affected by the over-stimulation (from 0 till to 100%) have been investigated by several simulations. Fig. 5 shows the firing series for the case of the Radiatum with an over-stimulation at 4 Hz. In this figure, the upper panel shows the basic condition while the other panels show the results of the over-stimulation for different percent-

two synaptic populations, while the structure, the position and the current characteristics of synapses did not vary, are shown in the bottom panel of the same figure. For this example, we have a mean spiking frequency of 1.05 Hz with a CV for ISIs of 0.91. The difference in the spiking activity was obtained by small variations of excitatory and inhibitory synaptic activation frequency (Fig. 3). In all the above cases, the CV of ISIs was close to the unit and within the range of values reported for in vivo recordings of cortical pyramidal neurons (Softky and Koch, 1993). Fig. 4 shows the membrane potential in the proximity of a spike generation. It has to be noted the large, irregular fluctuations of the membrane potential which occasionally can produce the threshold crossing and hence the firing of the neuron. This high irregularity is due to the contribution of all the excitatory and in-

Fig. 4. Membrane potential at the hillock for a time period encompassing the generation of a spike.

Fig. 5. Effects of synaptic stimulation on radiatum. The overstimulation – to 4 Hz – begins at 7 s as pointed by the arrow and stops at 8 s. The different panels show firing series for the following percentages of the stimulated synaptic population: 0% (A); 20% (B); 40% (C); 60% (D) ; 100% (E).

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Fig. 6. Effects of synaptic stimulation on radiatum. The firing frequency after the over-stimulation, averaged on a time period of 2 s, is shown for two stimulations at 3 or 4 Hz depending on the percentage of affected synaptic population.

ages of the excitatory synaptic population. A burst-like behavior appears evident from the panels related to larger fractions of stimulated synapses. The firing frequencies obtained by considering the number of spikes on a 2 s period immediately following the beginning of the overstimulation are reported in Fig. 6 for the two instances of 3 and 4 Hz depending on the percentage of the affected synaptic population. The two curves report results for just a sample. On the firing series to which results reported in Fig. 6 are related, some meaningful parameters have been computed such as mean ISI, CV and spiking frequency. Their course state is reported in Fig. 7. 5. Discussion In this paper, we presented a model of pyramidal neuron which accounts for many biological features. A nonsynchronous activation of synapses (produced by utilizing stochastic Poissonian processes) has been used to drive the neuronal activity. By combining the dis-

tance of each synapse with the (passive) cable properties of the dendrites, the contribution given by each filtered synaptic current to the membrane potential at the hillock has been computed. The stochastic fluctuations due to effects of the huge synaptic bombardment (also with low frequency of activation in each synapse) and of the variability of the peak amplitude of the currents at the synapses determine large fluctuations in the current arriving at the hillock as well as in the membrane potential at the same point. In fact, the membrane potential wanders from hyper-polarizing values up to the threshold value which is occasionally crossed (Figs. 3 and 4). The resulting randomness in the occurrence times of the spikes produces firing patterns whose variability, measured by CV and Fano factor, is comparable with that observed in in vivo experiments (Softky and Koch, 1993) and with that obtained by Shadlen and Newsome (1998) with a less complete model, but different from that computed by other computational methods (Stevens and Zador, 1998). However, it is better to stress that our model was investigated at low firing rates—about 1 Hz. This low level of the firing activity was obtained by activating at about 1 Hz the excitatory synapses and at about 10 Hz the inhibitory ones. Moreover, we have made a series of experiments where partial stimulation of the neuron is obtained by selectively increasing the firing frequency on the excitatory synapses in some modules of the neuron. In cortical pyramidal neurons, as well as in hippocampal neurons, this kind of stimulation is observed in vivo when inputs arrive to the neuron from different regions of the neocortex or other regions of brain. The different responses of our neuron model under different kinds of stimulation and for stimulation of different modules showed its usefulness to study the behavior and the coding ability of pyramidal neurons of the neocortex. Even though a lot of effort has been done to determine the geometrical structure of a pyramidal neuron and the disposition of all its synapses, the model is yet not complete in the present configuration. In fact, the active propagation along distal dendrites is lacking and also the shunting inhibition is not described. Moreover, the neuronal activity produced under variation of the input has to be more deeply investigated, also for the deepening of the hypothesis about a neuron as a coincidence detector. Moreover, we need to study the effects of feedback activity of inhibitory neurons in response to the variation of firing in the simulated pyramidal neuron. Modulation of synaptic activity from feedback inhibitory neurons, in fact, could play an important role in modifying the ISIs sequence. In conclusion, although the above results confirmed the robustness of our model, further investigations are needed for what concerns the under-

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Fig. 7. Effects of synaptic stimulation on radiatum. The two columns show values of mean ISI, CV and spiking frequency for stimulations at 3 or 4 Hz. On the horizontal axis the percentage of over-stimulated synaptic population is reported.

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