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Edited by: Matteo Barberis, University of Amsterdam, Netherlands

Reviewed by: Alexey Goltsov, Abertay University, United Kingdom; Frederic Von Wegner, Universitätsklinikum Frankfurt, Germany

*Correspondence: Tanguy Fardet

This article was submitted to Systems Biology, a section of the journal Frontiers in Neuroscience

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Experimental and numerical studies have revealed that isolated populations of oscillatory neurons can spontaneously synchronize and generate periodic bursts involving the whole network. Such a behavior has notably been observed for cultured neurons in rodent's cortex or hippocampus. We show here that a sufficient condition for this network bursting is the presence of an excitatory population of oscillatory neurons which displays spike-driven adaptation. We provide an analytic model to analyze network bursts generated by coupled adaptive exponential integrate-and-fire neurons. We show that, for strong synaptic coupling, intrinsically tonic spiking neurons evolve to reach a synchronized intermittent bursting state. The presence of inhibitory neurons or plastic synapses can then modulate this dynamics in many ways but is not necessary for its appearance. Thanks to a simple self-consistent equation, our model gives an intuitive and semi-quantitative tool to understand the bursting behavior. Furthermore, it suggests that after-hyperpolarization currents are sufficient to explain bursting termination. Through a thorough mapping between the theoretical parameters and ion-channel properties, we discuss the biological mechanisms that could be involved and the relevance of the explored parameter-space. Such an insight enables us to propose experimentally-testable predictions regarding how blocking fast, medium or slow after-hyperpolarization channels would affect the firing rate and burst duration, as well as the interburst interval.

Network bursting is an intermittent collective behavior that occurs spontaneously in neuronal populations. It is characterized by long quiet periods, with almost no spike emission, punctuated by brief periods of intense spiking activity, where the whole network displays high firing rates—most neurons emit at least 2 closely-packed spikes. This particular pattern is then repeated, with varying regularity, over long time intervals.

Such periodic and synchronized activity has been observed as an emergent phenomenon in large neuronal populations, both in brain regions (Meister et al.,

Recent experiments by Penn et al. (

Starting from these results and others (Ramirez et al.,

Let us insist on the fact that collective bursting, giving rise to “network bursts,” should not be confused with the individual behavior observed at the cellular level for “bursting” or “chattering” neurons. Though they share similar intervals of rapid firing followed by long quiet periods (Connors and Gutnick,

The periodic activity of the intrinsically oscillatory neurons present in culture populations and brain regions is assumed to rely on leak currents which affect their excitability (Suresh et al., _{Na, p} (Golomb et al., _{h} (Lüthi and McCormick, ^{+} (_{M}) current or the Ca^{2+} activated K^{+} currents (_{AHP}) (Sah and Louise Faber,

We show here that adaptive spiking is a sufficient condition for network bursting, confirming what was suggested by previous studies (Van Vreeswijk and Hansel,

We first describe the models used for the different units composing the system (neurons, synapses and network structure). Based on these, we derive an effective model which remains almost completely tractable, so that most of the properties of the collective dynamics can be predicted analytically. This model is based on successive approximations which were validated by numerical experiments: by dividing the cyclic behavior into several subdomains, we isolate regions where the activity can be solved under different approximations. The final solution is thus composed of the concatenation of these different approximations. We also used these simulations to verify and extend the predictions of our analytic equivalent model.

We chose the adaptive Exponential Integrate-and-Fire (aEIF) model (Brette and Gerstner,

where _{L} is its resting potential, _{th} is the threshold potential, _{r} is the reset potential. Ṽ_{peak} is the spike cutoff for the model. Ĩ_{e} is an external current to which the neuron can be submitted.

The main difference of this model compared to the well-known integrate-and-fire model is the presence of the second variable, the current _{s}, which is usually time dependent. The neuronal adaptation can be either subthreshold, through the coupling between Ṽ and

The exponential spike generation present in the aEIF model is more realistic than the hard threshold of the original Integrate-and-Fire model, which leads to unrealistically fast spiking during bursts. The soft threshold of the Izikevich model (Izhikevich, _{peak}) on the neuronal dynamics (Touboul,

In this study, and in accordance with the experimental observations for several types of pyramidal neurons, we use only neuronal parameters leading to adapting neurons which exhibit periodic spiking. This state is reached through the persistent current _{e}, which drives their progressive depolarization and makes them spike periodically; setting _{r} < _{th} ensures that the neurons are not intrinsically bursting, as described in Naud et al. (

Contrary to the resting state, where one stable and one unstable fixed point exist (points where both _{e} becomes high enough. In this spiking regime, no fixed point is present in phase space, which allows the neuron to depolarize until _{peak} before being reset to _{r}, thus following a discontinuous limit cycle.

Illustration of the resting and spiking behaviors can be found on Figures

For these parameter sets, we have

During the rest of the study, we use the dimensionless version of the model:

Details for the change of variables can be found in the first section of the Supplementary Material, “Neuronal model and parameters.” From then on, all equations involve only dimensionless variables and parameters.

The coupling strength between a pre-synaptic neuron _{s} transmitted from _{j}, the triggered PSC is felt by _{ji}, and is described by:

Where _{ji} is the strength of the synaptic connection from _{s} is the characteristic synaptic time, Θ(

This study is based on two non-spatial random network models: a fully homogeneous network with fixed in-degree which is useful to introduce the equivalent model, and more heterogeneous Gaussian in-degree networks which are supposed to be representative of connectivity in dissociated cultures (Cohen et al., _{i} (in-degree) of incoming connections originating from randomly chosen other neurons in the population. In the case of fixed in-degree networks, the in-degree _{i} is fixed and identical for each neuron. For Gaussian random networks, _{i} is drawn for each neuron from a Gaussian distribution with mean value _{k}. Note that the fixed-in-degree networks can be seen as the limit case of the Gaussian ones when the variance goes to zero. The out-degree distributions are binomial and identical in both cases. All networks where generated using the graph-tool or igraph backends of the NNGT library.

All transmissions between neurons in the network are subjected to the same delay

Where {_{j}} is the set of spike times for neuron

All dynamical simulations were performed using the NEST simulator (Kunkel et al.,

For each simulation we computed the average firing rate _{s} is the total number of spike and _{ν}, which would be the average interspike if the spikes were distributed uniformly. Considering _{ν}/2, 3

We derived an equivalent model that describes the system dynamics and predicts the range over which the characteristic frequencies can vary without the need to simulate the network dynamics.The model focuses on the fully synchronized dynamics, for which all neurons behave almost identically. The rationale of the model is most apparent if we first consider the case of a fixed in-degree network. As illustrated on Figure _{i}(_{j}(

Schematic representation of the equivalence between a fixed-in-degree network containing

This means that the network of

Based on this observation, exact for fixed in-degree networks, we propose a model of bursting dynamics for any synchronized network, where we describe the whole population through the behavior of an equivalent neuron, representative of the “average” dynamics. This neuron is subjected to the “average” input received by neurons in the network, and, under this simplified description, Equation (6) is now the same for every neuron in the network, since they are all approximated by this equivalent neuron. As they all receive the same number of spikes (_{s}) emitted at the same times {_{j}}, _{s}], and from the same number

This single dynamical system is then solved through several approximations depending on the network state. A typical approximation in the burst, on the interval [_{i}, _{i} + _{th}. On this interval, _{syn} = 0 and since _{w}, _{l}(_{l}(_{i}) = _{r} (see also Equations

For _{th}, _{peak}], we cannot solve the equation, but know from simulations that this simply leads the neuron to spike with a typical timescale of τ_{m} = 1.

From these analytic formula, we can then constrain the final solution through a self-consistent equation. The solution of the self-consistent equation will therefore assure that the spikes of one neuron during a burst sustain the burst itself and drive the subsequent ones (self-loop in the equivalent representation of Figure

This equivalent approach is applied here to three different synaptic models (instantaneous, continuous, and alpha-shaped synapses) leading to three transcendental self-consistent equations; details of mathematical developments can be found in the Supplementary Material. Python tools to solve the self-consistent equations and compute the characteristics of the bursting behavior are available on our GitHub repository; they are based on the

Thanks to the fast computation of the equivalent model, we were able to compute the dynamical properties for a large number of parameter sets. These results were normalized and analyzed through a Principal Component Analysis algorithm, using the

For each parameter set, we first ensure that there is no stable fixed-point in phase-space and that the model predicts a solution, i.e., the existence of bursts with mathematically coherent properties. Secondly, we assess the biological relevance of the solution by (1) ruling out dynamics for which the voltage decreases to values lower than −120 mV during the giant hyperpolarization following a burst; (2) restricting the maximum value of the slow current _{r} < _{th}—this restricts the neurons to single-spike intrinsic behaviors (Naud et al.,

These constraints limit the number of “valid” parameter sets and make the parameters inter-dependent; this leads to a non-trivial parameter/parameter correlation matrix (Figure

As mentioned in the introduction and discussed in the Supplementary Material, synchronization is highly resilient and we focus here solely on the fully synchronized bursting network. We start from individual neurons which are spiking periodically, a behavior that seems to originate from persistent sodium currents like _{Na, p} or _{h} in neuronal cultures (Penn et al., _{e}. When these neurons are coupled, however, their periodic dynamics is drastically modified as they adopt a collective bursting behavior (Borges et al.,

We describe the attractor characterizing the dynamics of the synchronous bursting state. Our key result details the properties of this attractor and shows how they are linked to both the biological parameters of the neurons and the network topology.

The behavior shows features of a relaxation oscillator (see ^{*}, which determines the burst termination and the start of a new cycle.

The main characteristic which determines the dynamics is the maximum value of the adaptation current, ^{*} reached at the end of a burst. It depends on the neuronal and network parameters, and qualitatively obeys the following equation (details in subsection 5.3 of the Supplementary Material):

where ^{*}, this equation directly shows that higher coupling (_{L}), or higher reset voltage (_{r}) will increase the bursting intensity. The effect of the transmission delay

Taking into account finer effects and spike-driven adaptation then leads to more complete equations delivering additional results about the influence of the remaining parameters. These are considered in more details in the Discussion section.

In the following subsections, we describe and explain the bursting dynamics, then discuss the more detailed, self-consistent versions of Equation (9) (complete derivation of these equations can be found in the Supplementary Material). Finally, we describe how our model accounts for the structural heterogeneity that is present in neuronal cultures.

The synchronous attractor is composed of intermittent bursts of activity, as shown in Figure

Attractors for three different networks of 1,000 identical neurons with average degree 100. Fixed-in-degree is represented by the blue solid line (spike positions are represented by empty squares and reset positions by full circles). For Gaussian in-degree networks, the logarithm of the number of states per bin—over 200 simulations with 4 cycles each—was used to compensate the non-constant velocity across the whole attractor. The larger attractor, in green, is associated to σ_{k} = 4; the smaller one, in purple, is for σ_{k} = 20; both attractors are delimited by a dashed line (limit of a unique visit per bin). Bin size is approximately 0.05 mV along the

This attractor is modified by the presence of heterogeneity in the network's topology—quantified by σ_{k} for Gaussian in-degree networks—which impacts both its duration and regularity. Indeed, heterogeneity noticeably smooths the average behavior and reduces the number of spikes in a burst which goes down from 6 spikes per burst for the fixed in-degree graph, to 3–5 if σ_{k} = 4, and is roughly reduced to 2 when σ_{k} = 20. For the fully synchronized fixed in-degree network, all neurons are responding to the exact same input—they receive spikes from the same number of neighbors—hence they are all equivalent to a single average neuron.

As can be seen on Figure ^{1}

Spike raster of bursting activity for a fixed 100-in-degree network. Inset provides details on the behavior of the neurons during a single burst, with successive synchronized burst slices separated by longer and longer intervals as the adaptation increases.

This inner structure, based on spike events, helps us define several quantities that characterize the dynamics such as the burst and inter-burst durations. However, information about the spike times alone is not sufficient to provide insights regarding the phenomena involved in the burst initiation or termination. Therefore, we will use the time evolution of the neuron's state variables to perform a phase-plane analysis and investigate possible mechanisms for both the bursting and recovery periods.

From the simulation, we can record the evolution of

_{r} and

The dynamics can be understood most easily when looking at _{min}—passing through points (1) to (4). At this point, the burst starts and ^{*}—point (5) on Figure

The evolution of _{e}, and the synaptic currents in the active period:

During the burst, each new spike induces a strong depolarization of the membrane, thus leading to another spike—point (4) to (5) on the figure.

Once ^{*}, its influence becomes predominant and prevents the neuron from firing; once the effect of the last spike vanishes, it drives a fast hyperpolarization of the neuron down to point (1).

After _{e}.

At this point, the potential starts increasing more rapidly as the first spike is initiated until the bursting starts again with (4), where the first spike predicted by the equivalent model occurs.

One of the main interests of this equivalent model is that it provides an intuitive understanding of the mathematical conditions describing the initiation and the termination of bursts. As shown on Figures _{min} represent the situation where the excitability of the neuron has become so high that it spontaneously emits a spike.

Trajectory of a “Dirac burst” in dimensionless phase space; the gray numbers indicate the order of the burst initiation. After a reset, the potential first decreases (leftmost parts of the trajectory) until the spike arrives (brown square), at which point the potential is suddenly shifted to the corresponding brown dot on the rightmost part of the trajectory. The decay before the spike arrival becomes more and more significant as ^{*}, denoted by the green dot, where the _{NV}(_{max} is reached (circled 0), the burst ends and the recovery period starts.

A key result is then the derivation of a condition for burst termination. We show that the end of the spiking sequence that constitutes a burst is ensured by the intrinsic dynamical properties of single neurons—through adaptation mechanisms—and does not require inhibition nor plastic synapses.

To understand the succession of spikes during the burst and why this spiking process comes to an end, we must introduce a description of the dynamic coupling between the neurons. We first explicit this coupling for two limit cases: firstly instantaneous couplings in perfectly regular fixed in-degree networks, using synapses modeled by Dirac delta functions (called Dirac synapses in the following); secondly, mimicking the effect of highly disordered networks, where synapses release a constant current over the entire burst duration. Thirdly, we consider a more biologically relevant coupling using alpha-shaped synapses, detailed in section 6 of the Supplementary Material, which lies between these two previous limits.

In general, the synaptic coupling _{s} between the neurons is time-dependent, which makes the resolution of the system's dynamics (Equation 3) highly complex. As a result the ^{*} in the case of the “alpha” synapses. Therefore, the Dirac and continuous synaptic models are more convenient since they enable us to get an insight on the bursting mechanisms through a static representation of the phase diagram during a burst.

The rationale for the condition of burst termination is most easily understandable in the case of regular networks assuming a coupling in the form of Dirac synapses. Indeed, the arrival of a spike then simply results in a step increment of the post-synaptic neuron's membrane potential:

where _{sp} is the time at which the spike is delivered to the post-synaptic neuron; _{s} is the total charge delivered by the spike and reflects the coupling strength in the network.

The behavior of the neuron can easily be understood by looking at the situation in phase space on Figure _{NV}(_{I}(_{m}, _{I}(_{w} (quasi-static approximation). Hence, either

Developing this condition mathematically leads to the following self-consistent equation:

where

For very heterogeneous networks, the broad in-degree distribution leads the neurons to fire at seemingly random times during the bursting period. In the limit where the time distribution of the spikes inside a burst becomes completely uniform, we can approximate it through a window-like synaptic current which is zero during the interburst, then jumps to a finite constant value during the burst.

To obtain an effect equivalent to the spikes described in the previous subsection, devoted the Dirac model, the total charge transmitted during the burst should be the same if an equal number of spikes is emitted. This condition reads, for an average in-degree

where _{s} is the number of spikes inside the burst. As described previously, the burst termination occurs when the trajectory crosses the

Trajectory of a burst in dimensionless phase space for neurons coupled via continuous synapses. Once the first spike occurs (marked by 4), the burst is initiated, i.e., a continuous current ^{*}, at _{max}, where it encounters the nullcline. This marks the end of the burst and the beginning of the recovery period (circled 0).

Because of the quasi-static hypothesis on _{NV}(

where

Once ^{*} has been computed using one of the theoretical models, we can derive all the dynamical properties, starting with:

where ⌈·⌉ denotes the ceiling function. Though the self-consistent equations derived above are less easy to interpret compared to the approximated solution (Equation 9), they allow precise quantitative predictions of the network's dynamics without too much computational cost.

Note that the neurons follow a well-defined and unique attractor, with _{max} is closer to the statistical value at which the neurons stop bursting: ^{*}.

The complete dynamics of the model can be completely captured by the relaxation behavior of

_{down} characterizes the time necessary for the neuron to undergo its strong hyperpolarization and reach its lowest membrane potential—from (0) to (1) on Figure

_{R} is the duration of the recovery—from (2) to (3),

_{fs} is the time necessary for the initiation of the first spike which is roughly equivalent to the membrane time constant τ_{m}—from (3) to (4).

This allows us to obtain the characteristic values of the dynamics (see section 8 of the Supplementary Material, “Resting period”, for detailed calculations):

_{s}

_{B} _{s}(

_{down}

_{R} ^{(2)} is the value of

_{down} + _{R} + 1.

Because these results are analytic, thus immediate to compute, this has the significant advantage over simulations that it allows us to quickly predict the properties of the collective dynamics for a large number of parameter sets, i.e., of individual neuron's behaviors.

In order to assess the separate influence of the different neuronal parameters on the bursting properties, we used the model to test in a systematic way the influence of the separate variables. As can be seen on Figure

Correlation matrix for the main characteristics of the bursting dynamics vs. neuronal parameters. 〈_{B}, _{s}, and 〈

This matrix allows us to confirm obvious trends, such as the negative influence of the driving current _{e} on the _{w} is almost linearly related to the _{r}) on the most visible features of the activity, namely the

Correlations for 〈_{s}) automatically increases 〈_{r} indeed reduces the interspike duration on the whole, the negative correlation between 〈_{s} as

Due to the sheer amount of calculation this would require, the theoretical values returned by the equivalent model during this large exploration of parameter-space cannot be verified by simulations in a systematic way. However, the distributions of the bursting characteristics (number of spikes, interburst, and burst duration) are in biologically relevant ranges—see Figure

Distributions of the burst burst duration _{B} and of the

Our description of periodic bursts predicts the main features of the synchronized bursting rhythmic activity such as its period and firing rate, which are significantly influenced by the presence of heterogeneity in the network's structure, as was already visible in Figure _{k}—increases, the sharpness of the synchronized burst slices decreases until the spikes contained in the burst become more uniformly distributed; this is clearly visible on Figure

Rasters of the bursting activity for 2 different Gaussian networks with 1,000 neurons and an average in-degree of 100; each inset details the inner structure of a burst with the successive slices. _{k} = 5 leads to well-defined synchronized burst slices inside the bursts. _{k} = 20 leads to fuzzy synchronized burst slices.

Our model is able to take this heterogeneity into account through three synaptic descriptions (Dirac, alpha-shaped, or “continuous”): this allows us to predict the interval in which the bursting properties of most networks should be contained. As shown on Figure

Variation of the _{k} = 4 (blue) and σ_{k} = 20 (green) are superimposed: the main curve represents the average value, while the filled area marks the 5th to 95th percentiles.

In all the simulations we performed, we observed that oscillating adaptive spiking neurons synchronize, then start emitting bursts of spikes as the coupling increases.

Our model provides a predictive framework which allows us to determine how this bursting behavior is affected by changes in the individual properties of the neurons.

In the following subsections, we first discuss the validity range of the analytic model. Then, through a thorough mapping of the aEIF parameters to ion channels and biological mechanisms, we make experimentally-testable predictions about the possible influence the main adaptation channels on the bursting behavior. Namely, we suggest how adaptation-channel blockers may affect the dynamics when applied on a bursting neuronal culture.

In order to get meaningful results within the framework of the present model, one must take care to use sets of parameters that lead to adaptive spiking neurons.

More importantly, the conceptual boundaries of the model are reached in the limit of either a very weakly or very strongly coupled neuronal network. For strong coupling the discrepancy between the equivalent model and the simulations mostly occurs because PSCs becomes so intense that a single input can generate several spikes. This can occur

Regarding the network structure, more heterogeneous (e.g., scale-free) networks may also be described by the “continuous-synapse” model on some range of the coupling strength as the qualitative bursting behavior is still present on such networks.

Despite its simplicity, the aEIF model takes into account most of the adaptation phenomena involved in biological neurons. Thus, voltage-gated subthreshold adaptation currents, like the muscarinic potassium current _{M} (Womble and Moises, _{r} in Equation (1). On an intermediate (“medium”) timescale, the current _{w}, which lengthens the effect of the potassium current after a burst. One of the limits of the model is its unique timescale for all of the adaptation-related features.

From the exploration of parameter-space, we obtain the correlation matrix of Figure ^{*}—and the spike-driven increment for the adaptation, ^{*}, which leads to longer interbursts. On the other hand, the quasi-static hypothesis states that the evolution of _{w}.

A significant advantage of this simple description is that the mechanisms proposed by our equivalent model, in light of the correlation matrix on Figure

Blocking the _{r}, hence increasing the number of spikes in a burst, leading to higher _{max}, therefore longer

Blocking the _{max} significantly, so it should not strongly impact the

Specific blocking of _{w}. In situation where adaptation has the strongest influence over the bursting period, this would lead to a significant decrease of the

These experiments would enable to test the adaptation hypothesis and assess the relative strength of the different processes we described. In fact, some previous studies by Empson and Jefferys (^{2+}-free medium to trigger the epileptiform activity. To assess the general validity of the proposed mechanisms, one would thus need additional measurements using cultures in physiological conditions, and where each ion-channel would be tested independently while recording larger fractions of the network, either through calcium imaging or MEAs.

Moreover, other features, such as slow modulation of extracellular potassium concentration due to neuronal activity (Bazhenov et al.,

Modification of the original dynamics (left), where only excitatory neurons are present, by the introduction of 20% of non-oscillating, fast-spiking inhibitory neurons (middle), or of plastic synapses exhibiting short-term depression (right). The coherence of the qualitative aspect over three very different systems is remarkable.

Eventually, previous studies (Cohen and Segal,

This study explains the dynamical processes determining synchronous network bursting of a population of oscillating neurons coupled through excitatory synapses. In particular we explain why adaptation is a sufficient condition for collective bursting. We reproduce a large range of biological rhythms with burst frequencies spanning almost 3 orders of magnitude, from a few hundred milliseconds to tens of seconds, in agreement with experimental observations.

Thanks to a phase-space analysis, we are able to propose a mechanism for the initiation and termination of the bursting period related to spike-driven adaptation, which we link to the underlying biological phenomena. The derivation of analytic equivalent models describing the complete bursting dynamics allows us to predict the evolution of the characteristics of the global behavior from the properties of the individual units—neurons and synapses. This enables us to propose a set of experiments which should clarify the role of adaptation currents in network bursting, as well as their relative importance compared to other biological processes such as exhaustion of vesicle pools.

In our description, each new spike in the burst is caused by the previous one, which means that the delay between the emission of a spike and its reception by the post-synaptic neuron has a significant influence on the dynamics. Indeed, we understand intuitively that the longer the delay, the lower the excitability of the neurons when the PSC arrives, since the membrane potential can decay to lower values. This fact, added to the effect of heterogeneity—which tends to reduce the interburst interval—hints at the existence of a limit to the spatial extension which can sustain coherent bursting. Exploring the effect of heterogeneity and spatial embedding (through propagation delays) therefore constitutes a natural continuation of this work. This is certainly necessary to address experimental observations in large cultures, such as the tendency of the activity to initiate in specific regions before it propagates to the rest of the network (Orlandi et al.,

Analytical study was conducted by MB and TF. TF also performed the numerical simulations and redaction of first draft. Additional simulations and analytical work were performed, respectively, by PM and SM. Final restructuring and adjustments were performed together by SB, TF, SM, and PM.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We would like to thank Elisha Moses, Yaron Penn, and Menahem Segal for their help and the invaluable discussions we had, as well as for sharing their data with us; Hans Ekkehard Plesser for his help and thorough reviews during the implementation and improvements of the numerical models in NEST; Sylvie Thérond for helping with the installation of NEST on the Turing supercomputer.

The Supplementary Material for this article can be found online at:

All codes used in this article are available on our GitHub repositories (

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^{2+}channels helps terminate epileptiform activity by activation of a Ca

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^{+}current to somatic bursting in CA1 pyramidal cells: combined experimental and modeling study

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^{+}currents in neurones: types, physiological roles and modulation

^{2+}channels involved in the generation of the slow afterhyperpolarization in cultured rat hippocampal pyramidal neurons

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^{1}Let us insist once again that the term