A dynamical computational model of theta generation in hippocampal circuits to study theta-gamma oscillations during neurostimulation

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This study presents a computational model to explore how neurostimulation could impact hippocampal theta oscillations. The computational model combines a detailed physiologically realistic hippocampus model and an abstract theta oscillator. The study could provide valuable predictions on pathological changes in this network. The modelling is based on convincing approaches that could be improved with experimental validation in future experiments.

https://doi.org/10.7554/eLife.87356.3.sa0

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Abstract

Neurostimulation of the hippocampal formation has shown promising results for modulating memory but the underlying mechanisms remain unclear. In particular, the effects on hippocampal theta-nested gamma oscillations and theta phase reset, which are both crucial for memory processes, are unknown. Moreover, these effects cannot be investigated using current computational models, which consider theta oscillations with a fixed amplitude and phase velocity. Here, we developed a novel computational model that includes the medial septum, represented as a set of abstract Kuramoto oscillators producing a dynamical theta rhythm with phase reset, and the hippocampal formation, composed of biophysically realistic neurons and able to generate theta-nested gamma oscillations under theta drive. We showed that, for theta inputs just below the threshold to induce self-sustained theta-nested gamma oscillations, a single stimulation pulse could switch the network behavior from non-oscillatory to a state producing sustained oscillations. Next, we demonstrated that, for a weaker theta input, pulse train stimulation at the theta frequency could transiently restore seemingly physiological oscillations. Importantly, the presence of phase reset influenced whether these two effects depended on the phase at which stimulation onset was delivered, which has practical implications for designing neurostimulation protocols that are triggered by the phase of ongoing theta oscillations. This novel model opens new avenues for studying the effects of neurostimulation on the hippocampal formation. Furthermore, our hybrid approach that combines different levels of abstraction could be extended in future work to other neural circuits that produce dynamical brain rhythms.

Introduction

Neurostimulation methods have emerged as promising therapeutic modalities to restore neurological functions in a broad range of motor and cognitive disorders (Gupta et al., 2023). In the context of learning and memory, deep brain stimulation (DBS) of the entorhinal area or hippocampus has been shown to either enhance (Suthana et al., 2012; Suthana and Fried, 2014; Titiz et al., 2017; Jun et al., 2019) or disrupt (Jacobs et al., 2016; Goyal et al., 2018; Lozano et al., 2016) memory encoding. These conflicting results may originate from differences in experimental protocols and from a poor understanding of their biophysical underpinnings. Among such mechanisms, the involvement of hippocampal theta oscillations (4–12 Hz) and their interactions with higher-frequency gamma oscillations (30–120 Hz) in memory-related processes has been reported in multiple studies (Lisman et al., 2005; de Almeida et al., 2007; Lega et al., 2012; Lin et al., 2017; Malkov et al., 2022; Abbaspoor et al., 2023). Moreover, the modulation of gamma oscillations by the phase of theta oscillations in hippocampal circuits, a phenomenon termed theta-gamma phase-amplitude coupling (PAC), correlates with the efficacy of memory encoding and retrieval (Jensen and Colgin, 2007; Tort et al., 2009; Canolty and Knight, 2010; Axmacher et al., 2010; Fell and Axmacher, 2011; Lisman and Jensen, 2013; Lega et al., 2016). Experimental and computational work on the coupling between oscillatory rhythms has indicated that it originates from different neural architectures and correlates with a range of behavioral and cognitive functions, enabling the long-range synchronization of cortical areas and facilitating multi-item encoding in the context of memory (Hyafil et al., 2015).

Neurostimulation protocols that affect these rhythms, such as theta burst stimulation (Titiz et al., 2017), have also been shown to optimally induce long-term potentiation (LTP) (Larson and Munkácsy, 2015). Another potential mechanism underlying the effect of hippocampal neurostimulation might be the reset of the phase of theta oscillations in response to exogenous inputs, such as a novel sensory input or a pulse of electrical stimulation applied to the fornix or perforant path (Buño et al., 1978; Williams and Givens, 2003). Theta phase reset is known to facilitate LTP (McCartney et al., 2004) and naturally occurs during both encoding and retrieval of associative memories (Kota et al., 2020). In this context, the design of a computational model that replicates memory-related theta-gamma oscillations and theta phase reset is of uttermost importance to investigate the effects of electrical stimulation on the hippocampal formation and possibly optimize neurostimulation protocols for memory improvement.

Models of memory-related theta-gamma oscillations in the hippocampal formation have been developed across different resolution levels, ranging from abstract mean-field approaches (Traub et al., 1997; Onslow et al., 2014; Segneri et al., 2020) to biophysically realistic conductance-based models (Lundqvist et al., 2006; Herman et al., 2013; Aussel et al., 2018). Neural masses, which represent the mean activity of a neuronal population, can generate gamma oscillations through reciprocal interactions between an excitatory and inhibitory population (Traub et al., 1997; Onslow et al., 2014), or even using a self-projecting inhibitory population (Segneri et al., 2020). Under excitatory oscillatory input at theta frequencies, these models are capable of generating theta-nested gamma oscillations. Similarly, these oscillations can be observed in more complex models of the hippocampal formation composed of single-compartment excitatory and inhibitory neurons connected through conductance-based synapses following the Hodgkin-Huxley formalism (Hodgkin and Huxley, 1952), and driven by a fixed oscillatory theta input (Aussel et al., 2018). Theta-nested gamma oscillations also appear in multi-compartment models of prefrontal cortex activity during memory retrieval (Lundqvist et al., 2006; Herman et al., 2013). Finally, several models have investigated the functional link between the hippocampal theta rhythm and memory processes, showing that encoding and retrieval occur at different phases within each theta cycle through phasic changes in neuronal activity and LTP (e.g. Hasselmo et al., 2002; Cutsuridis et al., 2010).

In terms of neurostimulation, most computational work has focused on the mechanisms underlying DBS of the basal ganglia for motor disorders such as Parkinson’s disease (Rubin and Terman, 2004; Pirini et al., 2009; Mina et al., 2013; Ebert et al., 2014), peripheral nerve stimulation (Rattay et al., 2003; Kipping and Nogueira, 2022), spinal cord stimulation (Rattay et al., 2000; Capogrosso et al., 2013), or has remained generic (Basu et al., 2018). However, models investigating neurostimulation of hippocampal circuits are scarce (Hendrickson et al., 2016; Bingham et al., 2018) and do not take into account the effects on theta-gamma oscillations. For example, a detailed multicompartment model of the rat dentate gyrus was able to replicate experimentally recorded local field potentials induced by different electrode placements and pulse amplitudes during stimulation of the perforant path (Bingham et al., 2018). To model the impact of neurostimulation on neuronal oscillations, a more abstract formalism based on Kuramoto phase oscillators (Kuramoto, 1984) has been introduced in the context of Parkinson’s disease and essential tremor, which enabled the design of novel neurostimulation paradigms that enhance or disrupt neuronal synchrony in basal ganglia circuits (Tass, 2003; Ebert et al., 2014; Asllani et al., 2018; Weerasinghe et al., 2019).

To our knowledge, there is currently no model of the hippocampal formation that is able to replicate both theta-gamma PAC and theta phase reset during neurostimulation. To investigate the effects of neurostimulation on hippocampal circuits while taking into account these two mechanisms, we modified an existing biophysical model of the hippocampal formation (Aussel et al., 2018). In this original model, however, the theta rhythm was considered as an oscillatory input of fixed amplitude and phase velocity, which is inconsistent with theta phase reset. To circumvent this limitation, we combined this biophysical model with abstract Kuramoto oscillators that acted as a dynamical source of theta rhythm, thereby modeling medial septum inputs to the hippocampal formation. This new hybrid dynamical model could generate both theta-nested gamma oscillations and theta phase reset, following a particular phase response curve (PRC) inspired by experimental literature (Lengyel et al., 2005; Akam et al., 2012; Torben-Nielsen et al., 2010).

We then leveraged this model to explore the effect of single-pulse and pulse-train stimulation on theta-gamma oscillations. In the absence of theta input from the medial septum, single-pulse stimulation produced a transient effect consisting of one or several bursts of activity, depending on stimulation amplitude. The presence of multiple bursts depended on the single-cell calcium dynamics and M-type potassium adaptation current. In the presence of weak theta input, designed to mimic a pathological state that impairs theta-gamma oscillations, single-pulse stimulation could produce long-lasting or even persistent activity by switching the network to a highly synchronized state characterized by theta-nested gamma oscillations. When phase reset was not included in the model, this effect was more pronounced when the stimulation pulse was delivered at the peak of the theta rhythm. However, when strong theta reset was considered, the phase at which stimulation was delivered did not influence the outcome. In the presence of an even weaker theta input, mimicking a pathological state that completely abolishes theta-gamma oscillations, only pulse train stimulation could restore physiological theta-gamma oscillations during the stimulation period. As for the previous results, this effect was phase-dependent only when theta reset was not included.

These results provide a new framework to interpret neurostimulation interventions that interfere with hippocampal oscillations and aim at improving memory function. It can be further extended to investigate the effects of more complex neurostimulation protocols, and the impact of stimulation location and amplitude on the observed network dynamics.

Results

A computational model of the hippocampal formation with dynamical theta input

We first developed a computational model of hippocampal circuits, able to generate both theta-nested gamma oscillations and theta phase reset. To achieve this, we combined an existing conductance-based model of the hippocampal formation (Aussel et al., 2018) with a set of Kuramoto phase oscillators (Kuramoto, 1984) that were used to model the dynamical theta input originating from pacemaker neurons in the medial septum (Wang, 2002, Figure 1A). Such ensembles of oscillators are designed to exhibit a strong phase reset of their collective rhythm in response to a perturbation (Levnajić and Pikovsky, 2010). The hippocampal model contained excitatory and inhibitory neuronal populations in the entorhinal cortex (EC), dentate gyrus (DG), CA3 and CA1 fields within a coronal slice of the human hippocampus (Figure 1B–C). The output of the hippocampal formation (i.e. the activity of CA1 pyramidal neurons) was provided as an input to the Kuramoto oscillators, simulating the hippocampal-septal projections through the fornix (Williams and Givens, 2003; Nuñez and Buño, 2021; Takeuchi et al., 2021). The oscillations produced by the collective behavior of Kuramoto oscillators represented a population average of their activity: highly synchronized and desynchronized states generated respectively high and low amplitude theta oscillations (Figure 1D). The number of oscillators was chosen so that their synchronization level (i.e. their order parameter) and their frequency distribution were sufficiently close to their asymptotic behavior for a large number of oscillators (Figure 1—figure supplement 1).

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Dynamical computational model of the hippocampal formation and medial septum oscillatory drive.

(A) Anatomical representation of the neuronal types and interconnections within and between the medial septum and the hippocampal formation (EC: entorhinal cortex, DG: dentate gyrus, CA3 and CA1 fields of the hippocampus). (B) Simplified anatomy of the hippocampal formation, modeled as a 15-mm-thick cylindrical slice, with spatially segregated excitatory and inhibitory neurons (blue: excitatory neurons, consisting of granule cells in DG and pyramidal cells in other areas; red: inhibitory basket cells). (C) Model architecture and connectivity. Each area is comprised of one excitatory and one inhibitory neuronal population. Theta drive is provided through input from the medial septum, which is modeled as a set of 250 Kuramoto oscillators and receives feedback connections from CA1. Electrical stimulation Is modeled as an intracellular current affecting both excitatory and inhibitory populations in the targeted area (shown here for CA1). (D) Illustration of Kuramoto oscillators with two different levels of synchronization. Each dot represents one oscillator, its position on the circle indicates its phase and its color its angular velocity. Higher synchronization corresponds to a clustering of the dots around a similar phase. ‘ $r$ ’ indicates the order parameter, which is a measure of synchronization.

Biologically, GABAergic neurons from the medial septum project to the EC, CA3, and CA1 fields of the hippocampus (Tóth et al., 1993; Hajós et al., 2004; Manseau et al., 2008; Hangya et al., 2009; Unal et al., 2015; Müller and Remy, 2018). Although the respective roles of these different projections are not fully understood, previous computational studies have suggested that the direct projection from the medial septum to CA1 is not essential for the production of theta in CA1 microcircuits (Mysin et al., 2019). Since our modeling of the medial septum is only used to generate a dynamic theta rhythm, we opted for a simplified representation where the medial septum projects only to the EC, which in turn drives the different subfields of the hippocampus. In our model, Kuramoto oscillators are therefore connected to the EC neurons and they receive projections from CA1 neurons (see Materials and methods for more details).

From a conceptual point of view, our model is thus composed of excitatory-inhibitory (E-I) circuits connected in series, with a feedback loop going through a population of coupled phase oscillators. In the next sections, we first describe the generation of gamma oscillations by individual E-I circuits (Figure 2), and illustrate their behavior when driven by an oscillatory input such as theta oscillations (Figure 3). We then present a thorough characterization of the effects of theta input and stimulation amplitude on theta-nested gamma oscillations (Figure 4 and Figure 5). Finally, we present some results on the effects of neurostimulation protocols for restoring theta-nested gamma oscillations in pathological states (Figure 6 and Figure 7).

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Emergence of gamma oscillations in coupled excitatory-inhibitory populations under ramping input to both populations.

(A) Two coupled populations of excitatory pyramidal neurons ( $N_{E}$ = 1000) and inhibitory interneurons ( $N_{I}$ = 100) are driven by a ramping current input (0 nA to 1 nA) for 5 s. As the input becomes stronger, oscillations start to emerge (shaded green area), driven by the interactions between excitatory and inhibitory populations. The green inset shows the raster plot (neuronal spikes across time) of the two populations during the green shaded period (red for inhibitory; blue for excitatory). When the input becomes sufficiently strong (shaded magenta area), the populations become highly synchronized and produce oscillations in the gamma range (at approximately 50 Hz). The spectrogram (bottom panel) shows the power of the instantaneous firing rate of the pyramidal population as a function of time and frequency. It reveals the presence of gamma oscillations that emerge around 2 s and increase in frequency until 4 s, when they settle at approximately 60 Hz. (B) Similar depiction as in panel A. with the pyramidal-interneuronal populations decoupled. The absence of coupling leads to the abolition of gamma oscillations, each cell spiking activity being driven by its own inputs and intrinsic properties.

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Theta-nested gamma oscillations and theta phase reset in response to a single stimulation pulse.

(A) Representative example of the network behavior, spontaneously producing theta-nested gamma oscillations and characterized by a reset of the theta phase following a single stimulation pulse (vertical grey line, applied here in CA1 at a theta phase of $π / 2$ , that is in the middle of the descending slope). Top to bottom: theta rhythm originating from the medial septum and provided as an input to the EC ( $f_{θ}$ : mean oscillation frequency), instantaneous phase of the theta rhythm, raster plots indicating the spiking activity of CA1 excitatory (blue) and inhibitory (red) neurons ( $μ_{E}$ and $μ_{I}$ : mean firing rates within the shaded area), average population firing rates in CA1 (computed as a windowed moving average with a sliding window of 100ms with 99% overlap), spectrograms for each CA1 population (windowed short-time fast Fourier transform using a Hann sliding window: 100ms with 99% overlap). Spectrograms show gamma oscillations (around 60 Hz) modulated by the underlying theta rhythm (∼4 Hz), indicating theta-gamma PAC. Theta phase reset after stimulation is associated with a rebound of spiking activity and theta-nested gamma oscillations. (B) Power spectral densities of the CA1 firing rates. Theta peaks are found at 4 Hz for excitatory and inhibitory cells. Gamma activity is located between 40 and 80 Hz. (C) PAC as a function of theta phase and gamma frequency. The polar plot represents the amplitude of gamma oscillations (averaged across all theta cycles, see Materials and methods) at each phase of theta (theta range: 3–9 Hz, phase indicated as angular coordinate) and for different gamma frequencies (radial coordinate, binned in 10 Hz ranges), indicating that gamma oscillations between 40 and 80 Hz occur preferentially around the peak of theta. The MI gives an overall quantification of how the phase of low-frequency oscillations (3–9 Hz) modulates the amplitude of higher-frequency oscillations (40–80 Hz) (see Materials and methods and Figure 3—figure supplement 2). (D) PRC in response to a single stimulation pulse applied in CA1 at various phases of the ongoing theta rhythm and for various stimulation amplitudes (color-coded). The phase difference (left y-axis) shows the theta phase induced by the stimulation pulse (computed 2.5ms after the pulse), compared to the phase computed at the same time in a scenario without stimulation. Positive and negative phase differences indicate phase advances and delays, respectively. The grey trace shows the normalized amplitude of theta (right $y$ axis) for different phases, used to indicate the peak and trough of the rhythm. Stimulation applied in the ascending slope of theta ( $[- π, 0]$ ) produced a phase advance and accelerated the rhythm towards its peak (0 radians). Conversely, stimulation during the descending slope ( $[0, π]$ ) produced a phase delay that slowed down the rhythm. Higher stimulation amplitudes yielded a stronger effect.

Figure 4

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CAN and adaptation (M) currents modulate neuronal responses to single-pulse stimulation in the absence of theta input.

(A) Network response to single-pulse stimulation in the absence of medial septum input. Stimulation (grey vertical line, 10 nA, applied in CA1) induced an instantaneous burst of activity lasting about 20ms in both excitatory and inhibitory CA1 neurons, followed by a secondary burst approximately 200 ms later (raster plot and firing rate traces), associated with specific CAN- and M- currents dynamics (bottom traces, illustrated for a representative CA1 excitatory neuron). Positive ( $I_{M}$ ) and negative ( $I_{C A N}$ ) currents indicate respectively a hyperpolarizing and depolarizing effect on the cell membrane potential. (B) Similar representation as in A., but in the absence of CAN channel. Stimulation induced only a single burst of activity, indicating that CAN channels are necessary to observe a rebound of activity. (C) Number of bursts in CA1 spiking activity following a single stimulation pulse at various amplitudes (x-axis), shown both in the presence and absence of CAN channels in excitatory neurons. The absence of the CAN current leads to the abolition of the second burst, irrespective of stimulation amplitude.

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Medial septum oscillatory drive and stimulation amplitude govern the steady-state response to single-pulse stimulation.

(**A-C**) Network responses to single-pulse stimulation (vertical line) under medial septum input (with phase reset), shown for different amplitudes of the medial septum oscillatory drive (A-C: increasing oscillator amplitudes). (A) Low oscillatory input: stimulation induces only two bursts of spiking activity as in Figure 4. (B) Medium oscillatory input: a single stimulation pulse switches network behavior from no activity to sustained oscillations driven by the medial septum. (C) Higher oscillatory input: theta drive is capable of inducing self-sustained theta-nested gamma oscillations. In this case, stimulation is delivered at the peak of theta oscillations and does not show a pronounced effect on theta-gamma oscillations. (D) Steady-state response to single-pulse stimulation as a function of medial septum oscillatory input (x-axis) and stimulation amplitude (y-axis), characterized by three metrics: theta power (3–9 Hz), gamma power (40–80 Hz), and PAC (quantified using the MI). White dots: parameter combinations corresponding to panels A-C.

Figure 6

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Single-pulse stimulation phase differentially affects network responses depending on the presence of theta phase reset.

All results shown here were obtained for parameters from Figure 5B (theta oscillation amplitude: 0.13 nA, stimulation amplitude: 7.0 nA). A single stimulation pulse was delivered at the peak (**A, B**) or trough (**C, D**) of the underlying theta rhythm, either in the presence (**A, C**) or absence (**B, D**) of theta phase reset. With phase reset, both peak and trough stimulation switch network behavior from no activity to sustained oscillations. Without phase reset, only peak stimulation can induce sustained oscillations. (E). Quantification of theta power, gamma power, and PAC (measured using the MI) in CA1 excitatory (blue) and inhibitory (red) populations in all four cases (metrics are computed in the shaded areas of panels A-D).

Figure 7

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Pulse train stimulation restores theta-nested gamma oscillations depending on stimulation timing and theta phase reset.

All results shown here were obtained for parameters from Figure 5A (theta oscillation amplitude: 0.05 nA; stimulation amplitude: 7.0 nA). Representations are similar to Figure 6, with the difference that stimulation consisted of a pulse train delivered at 6 Hz for a duration of 2 s (individual pulses indicated by grey dots, first pulse by a triangle). The pulse train was delivered at the peak (**A, B**) or trough (**C, D**) of the underlying theta rhythm, either in the presence (**A, C**) or absence (**B, D**) of theta phase reset. With phase reset, both peak and trough stimulation switch network behavior from no activity to sustained oscillations. Without phase reset, only peak stimulation can induce sustained oscillations. E. Quantification of theta power, gamma power, and PAC (measured using the MI) in CA1 excitatory (blue) and inhibitory (red) populations in all four cases (metrics are computed during the pulse train, within the shaded areas of panels A-D).

Generation of gamma oscillations by E-I circuits

It is well established that a network of interconnected pyramidal neurons and interneurons can give rise to oscillations in the gamma range, a mechanism termed pyramidal-interneuronal network gamma (PING) (Traub et al., 2004; Onslow et al., 2014; Segneri et al., 2020). This mechanism has been observed in several optogenetic studies with gradually increasing light intensity (i.e. under a ramp input) affecting multiple different circuits, such as layer 2–3 pyramidal neurons of the mouse somatosensory cortex (Adesnik and Scanziani, 2010), the CA3 field of the hippocampus in rat in vitro slices (Akam et al., 2012), and in the non-human primate motor cortex (Lu et al., 2015). In all cases, gamma oscillations emerged above a certain threshold in terms of photostimulation intensity, and the frequency of these oscillations was either stable or slightly increased when increasing the intensity further. We sought to replicate these findings with our elementary E-I circuits composed of single-compartment conductance-based neurons driven by a ramping input current (Figure 2 and Figure 2—figure supplement 1). As an example, all the results in this section will be shown for an E-I circuit that has similar connectivity parameters as the CA1 field of the hippocampus in our complete model (see section ‘Hippocampal formation: inputs and connectivity’ in the Materials and methods).

For low input currents provided to both neuronal populations, only the highly excitable interneurons were activated (Figure 2A). For a sufficiently high input current (i.e. a strong input that could overcome the inhibition from the fast-spiking interneurons), the pyramidal neurons started spiking as well. As the amplitude of the input increased, the activity of both neuronal populations became synchronized in the gamma range, asymptotically reaching a frequency of about 60 Hz (Figure 2A bottom panel). Decoupling the populations led to the abolition of gamma oscillations (Figure 2B), as neuronal activity was determined solely by the intrinsic properties of each cell. Interestingly, when the ramp input was provided solely to the excitatory population, we observed that the activity of the pyramidal neurons preceded the activity of the inhibitory neurons, while still preserving the emergence of gamma oscillations (Figure 2—figure supplement 1A). As expected, decoupling the populations also abolished gamma oscillations, with the excitatory neurons spiking a frequency determined by their intrinsic properties and the inhibitory population remaining silent (Figure 2—figure supplement 1B).

To further characterize the intrinsic properties of individual inhibitory and excitatory neurons, we derived their input-frequency (I-F) curves, which represent the firing rate of individual neurons in response to a tonic input (Figure 2—figure supplement 2A). We observed that for certain input amplitudes, the firing rates of both types of neurons was within the gamma range. Interestingly, in the absence of noise, each population could generate by itself gamma oscillations that were purely driven by the input and determined by the intrinsic properties of the neurons (Figure 2—figure supplement 2B). Adding stochastic Gaussian noise in the membrane potential disrupted these artificial oscillations in decoupled populations (Figure 2—figure supplement 2C). All subsequent simulations were run with similar noise levels to prevent the emergence of artificial gamma oscillations.

Another potent way to induce gamma oscillations is to drive fast-spiking inhibitory neurons using pulsed optogenetic stimulation at gamma frequencies, a strategy that has been used both in the neocortex (Cardin et al., 2009) and hippocampal CA1 (Iaccarino et al., 2016). In particular, Cardin and colleagues systematically investigated the effect of driving either excitatory or fast-spiking inhibitory neocortical neurons at frequencies between 10 and 200 Hz (Cardin et al., 2009). They showed that fast-spiking interneurons are preferentially entrained around 40–50 Hz, while excitatory neurons respond better to lower frequencies. To verify the behavior of our model against these experimental data, we simulated pulsed optogenetic stimulation as an intracellular current provided to our reduced model of a single E-I circuit. Stimulation was applied at frequencies between 10 and 200 Hz to excitatory cells only, to inhibitory cells only, or to both at the same time (Figure 2—figure supplement 3). The population firing rates were used as a proxy for the local field potentials (LFP), and we computed the relative power in a 10 Hz band centered around the stimulation frequency, similarly to the method proposed in Cardin et al., 2009. When presented with continuous stimulation across a range of frequencies in the gamma range, interneurons showed the greatest degree of gamma power modulation (Figure 2—figure supplement 3). Furthermore, when the stimulation was delivered to the excitatory population, the relative power around the stimulation frequency dropped significantly in frequencies above 10 Hz, similar to the reported experimental data (Cardin et al., 2009). The main difference between our simulation results and these experimental data is the specific frequencies at which fast-spiking interneurons showed resonance, which was around 40 Hz in the mouse barrel cortex and around 90 Hz in our model, a fast gamma rhythm. This could be attributed to several factors, such as differences in the cellular properties between cortical and hippocampal fast-spiking interneurons, or the differences between the size of the populations and their relevant connectivity in the cortex and the hippocampus.

Theta-gamma oscillations and theta phase reset under dynamical theta input

Once we validated that our elementary E-I circuits were able to generate gamma oscillations under an input of sufficiently high amplitude, we studied the behavior of the whole model when the theta input was dynamically provided by the Kuramoto oscillators as described above (Figure 1). As in the original model (Aussel et al., 2018), the input theta rhythm drove the network to produce spiking activity in the gamma range preferentially around the peak of theta in all excitatory and inhibitory populations (Figure 3A and Figure 3—figure supplement 1). Spectrograms of the CA1 population firing rates revealed that these bursts of activity around each theta peak were characterized by oscillations around 60 Hz (Figure 3A). This was confirmed by power spectral densities, which showed a clear peak at 4 Hz (corresponding to the theta drive) and increases in gamma band activity between 40 and 80 Hz (Figure 3B). To quantify PAC between theta and gamma oscillations, we used the modulation index (MI) (Tort et al., 2008; Tort et al., 2010), which has been shown to outperform other similar measures (Hülsemann et al., 2019). However, we discovered that this metric would give erroneous results in our simulated datasets due to the absence of certain frequency components. To overcome this limitation and avoid artifacts, uniform noise was added to the firing rates prior to computing the MI (Figure 3—figure supplement 2). We first visualized the quantification of PAC using the comodulogram, which indicates the MI as a function of two frequencies corresponding to the modulating phase signal and the modulated amplitude signal. This analysis confirmed that gamma-band signals between 45 and 75 Hz were modulated by lower theta frequencies between 3 and 6 Hz, which we quantified by computing a global MI in frequency ranges encompassing theta (3–9 Hz) and gamma (40–80 Hz) (Figure 3—figure supplement 2B). To refine this analysis, we also computed a similar MI between the amplitude of various gamma frequencies and the phase of theta, which indicated that oscillations between 40 and 80 Hz occur preferentially around the peak of theta (Figure 3C and Figure 3—figure supplement 2B).

Next, we verified that our model was able to display theta phase reset during single-pulse stimulation (depolarizing pulse, 1 ms duration). This mechanism is tightly linked with the concept of PRC, which characterizes the phase delay or advancement that follows a single pulse delivered to an oscillatory system, as a function of the phase at which this input is delivered. Although there is no direct measurement of the PRC of septal neurons, such characterizations have been performed for individual pyramidal cells in the CA3 and CA1 fields of the hippocampus (Lengyel et al., 2005; Kwag and Paulsen, 2009; Akam et al., 2012). These PRCs appear biphasic and show a phase advancement (respectively delay) for stimuli delivered in the ascending (respectively descending) slope of theta. We modeled this behavior by a specific term (which we called the phase response function) in the general equation of the Kuramoto oscillators (see methods, Equation 1). Importantly, introducing a phase offset in the phase response function disrupted theta-nested gamma oscillations (Figure 3—figure supplement 3), which suggests that the septohippocampal circuitry must be critically tuned to be able to generate such oscillations. The strength of phase reset could also be adjusted by a gain that was manually tuned. In the presence of the physiological phase response function and of a sufficiently high reset gain, a single stimulation pulse delivered to all excitatory and inhibitory CA1 neurons could reset the phase of theta to a value close to its peaks (Figure 3A). We computed the PRC of our simulated data for different stimulation amplitudes and validated that our neuronal network behaved according to the phase response function set in our Kuramoto oscillators (Figure 3D). It should be noted that including this phase reset mechanism affected the generated theta rhythm even in the absence of stimulation, extending the duration of the theta peak and thereby slowing down the frequency of the generated theta rhythm.

Importantly, our approach is generalizable and can be applied to other models producing theta-nested gamma oscillations. For instance, we adapted the neural mass model by Onslow and colleagues (Onslow et al., 2014), replaced the fixed theta input by a set of Kuramoto oscillators, and demonstrated that it could also generate theta phase reset in response to single-pulse stimulation (Figure 3—figure supplement 4). These results illustrate that the general behavior of our model is not specific to the tuning of individual parameters in the conductance-based neurons, but follows general rules that are captured by the level of abstraction of the Kuramoto formalism.

Overall, we successfully developed a new model of the hippocampal formation able to exhibit both theta-nested gamma oscillations and theta phase reset in response to stimulation. We then decided to explore further the effects of various stimulation protocols on its dynamics.

Effects of theta input and stimulation amplitudes on theta-gamma oscillations

We investigated the behavior of the model across multiple states of varying septal theta input amplitude in response to single-pulse stimulation delivered to CA1. In the absence of theta drive, a single stimulation pulse elicited either zero or two bursts of spiking activity (depending on stimulation amplitude) separated by about 200 ms (Figure 4A and C). We first sought to understand the origin of this second burst, as it showed that even a single pulse could induce transient periodic activity around 5 Hz in our model, a frequency within the theta range. A previous model has shown that the presence of Calcium-Activated Non-specific cationic (CAN) currents can lead to self-sustained theta oscillations in the hippocampus (Giovannini et al., 2017). Moreover, this study showed a direct link between the increased excitation provided by the CAN current and the spike-frequency adaptation properties of the M current, directly affecting transitions from an asynchronous low-firing regime to synchronous bursting. We tested the role of the CAN current in the response to single-pulse stimulation by completely removing it from our simulations, which abolished the second burst of activity (Figure 4B). Moreover, the time interval separating the two bursts likely resulted from the interplay between the depolarizing CAN current and hyperpolarizing M current (Figure 4A).

In the presence of dynamic theta input, the effects of single-pulse stimulation depended both on theta input amplitude and stimulation amplitude, highlighting different regimes of network activity (Figure 5 and Figure 5—figure supplement 1, Figure 5—figure supplement 2, Figure 5—figure supplement 3). For low theta input, theta-nested gamma oscillations were initially absent and could not be induced by stimulation (Figure 5A). At most, the stimulation could only elicit a few bursts of spiking activity that faded away after approximately 250ms, similar to the rebound of activity seen in the absence of theta drive. For increasing theta input, the network switched to an intermediate regime: upon initialization at a state with no spiking activity, it could be kicked to a state with self-sustained theta-nested gamma oscillations by a single stimulation pulse of sufficiently high amplitude (Figure 5B). This regime existed for a range of septal theta inputs located just below the threshold to induce self-sustained theta-gamma oscillations without additional stimulation, as characterized by the post-stimulation theta power, gamma power, and theta-gamma PAC (Figure 5D). Removing CAN currents from all areas of the model abolished this behavior (Figure 5 - figure supplement 4), which is interesting given the role of this current in the multistability of EC neurons (Egorov et al., 2002; Fransén et al., 2006) and in the intrinsic ability of the hippocampus to generate theta-nested gamma oscillations (Giovannini et al., 2017). For the highest theta input, the network became able to spontaneously generate theta-nested gamma oscillations, even when initialized at a state with no spiking activity and without additional neurostimulation Figure 5C.

Neurostimulation for restoring theta-gamma oscillations in pathological states

Based on the above analyses, we considered two pathological states: one with a moderate theta input (i.e. moderately weak projections from the medial septum to the EC) that allowed the initiation of self-sustained oscillations by single stimulation pulses (Figure 5, point B), and one with a weaker theta input characterized by the complete absence of self-sustained oscillations even following transient stimulation (Figure 5, point A). In each case, we sought to assess whether single-pulse or pulse train stimulation could induce or restore theta-nested gamma oscillations and whether this effect depended on the phase at which stimulation was delivered (i.e. at the peak or trough of the theta cycle). We hypothesized that any possible phase relationship would also depend on the phase reset mechanism. To test this hypothesis, we ran a series of simulations using two different models: one without phase reset and one with strong phase reset (i.e. the reset gain was set at the value used in Figure 3).

In the case of a moderate theta input and in the presence of phase reset, delivering a pulse at either the peak or trough of theta could induce theta-nested gamma oscillations (Figure 6A and C). By contrast, in the absence of phase reset, only stimulation delivered at the peak of theta was able to induce such oscillations, after some time delay (Figure 6B and D). Quantification of these results in terms of theta power, gamma power, and theta-gamma PAC showed strongly similar responses between stimulation delivered at the peak or trough of theta with phase reset enabled, similar but weaker responses with stimulation delivered at the peak of theta with phase reset disabled, and no response in the case of trough stimulation with phase reset disabled (Figure 6E).

In the case of a weak theta input that completely abolished neuronal oscillations, we delivered stimulation pulses continuously at a frequency matching that of the underlying theta rhythm for a duration of 2 s, in order to restore physiological oscillations (Figure 7). The stimulation onset was timed to either the peak or trough of the ongoing theta cycle. The continuous delivery of pulses produced similar results as single-pulse stimulation. With phase reset, pulse train stimulation restored theta-nested gamma oscillations within the whole network, irrespective of the phase of stimulation onset (Figure 7A and C). Notably, stimulation delivered at the trough forced a reset of the phase of theta rhythm and led to the subsequent delivery of pulses at the peak of theta. Interestingly, in the absence of phase reset, peak-targeted stimulation induced theta-gamma oscillations in all network areas easily, while trough-targeted stimulation created artificial bursts that propagated in other areas with difficulty, requiring multiple pulses to achieve a fraction of the results of peak-targeted stimulation (Figure 7B and D). Comparing these simulations based on theta power, gamma power, and theta-gamma PAC within CA1 (Figure 7E) showed similar albeit less striking differences as single-pulse stimulation, possibly because CA1 was driven directly by the stimulation. Notably, gamma power in the absence of theta phase reset, was higher when utilizing pulse trains. These differences were even more pronounced in areas other than CA1, as can be seen by the gradual emergence of oscillations for trough stimulation without phase reset (Figure 7D).

Discussion

Highlights

In summary, we have developed a novel computational model to investigate the effects of electrical stimulation on a slice of the hippocampal formation, incorporating two important features related to memory: theta-nested gamma oscillations and theta phase reset. The key innovation compared to previous models (e.g. Aussel et al., 2018) is the introduction of a set of abstract Kuramoto oscillators, which represent pacemaker neurons in the medial septum and are interfaced with biophysically realistic neuronal models in the hippocampus. From a methodological point of view, this hybrid interfacing between two levels of abstraction represents an innovation in itself and could be applied to other systems or brain structures that are driven by dynamical rhythms. The main outcomes reported here relate to the importance of the theta reset mechanism when examining the effects of neurostimulation on hippocampal oscillations.

A very interesting finding concerns the behavior of the model in response to single-pulse stimulation for certain values of the theta amplitude (Figure 5). For low theta amplitudes, a single stimulation pulse was capable of switching the network behavior from a state with no spiking activity to one with prominent theta-nested gamma oscillations. Whether such an effect can be induced in vivo in the context of memory processes remains an open question. Nevertheless, delivering a single stimulation pulse bilaterally to the human hippocampus during a memory task is sufficient to impair memory encoding (Lacruz et al., 2010), suggesting that even single-pulse stimulation can indeed have wide network effects that are behaviorally relevant.

The second main finding is that the timing of individual stimulation pulses with respect to the phase of the ongoing theta rhythm matters differently depending on the presence or absence of phase reset (Figure 6 and Figure 7). Human intracranial stimulation data indicate that the receptivity of hippocampal circuits to single-pulse stimulation is modulated by the phase of theta (Lurie et al., 2022). A number of studies have also reported conflicting results in terms of memory outcomes, which could potentially be attributed to the induction of phase reset through stimulation (Suthana et al., 2012; Jacobs et al., 2016). To our knowledge, however, the degree of phase reset that follows each stimulation pulse remains unknown and should be investigated in future experimental studies. From a technological point of view, the two regimes with and without phase reset have opposite predictions concerning the need for closed-loop stimulation protocols that would trigger stimulation in real-time based on the phase of ongoing theta oscillations. Such phase-triggered stimulation would be most useful if the phase reset mechanism remains relatively limited. When this mechanism becomes too strong, regular continuous stimulation appears sufficient to restore physiological theta-gamma oscillations (Figure 7).

It should also be noted that the reset gain in our simulations (Equation 1, gain $G_{r e s e t}$ ) was either completely turned off or set at a high value, producing a strong effect that always reset the phase of theta to its peak. In reality, the degree of theta phase reset is dynamic, depending on the environment and associated task requirements (Rizzuto et al., 2003; Mormann et al., 2005; Jackson et al., 2008), and may be affected by neurodegenerative disorders that affect the connections between the hippocampal formation and the medial septum. Importantly, our proposed framework can simulate intermediate values of the reset gain, which should ideally be fitted to experimental data in future applications.

Finally, we modeled pathological states by reducing the maximum amplitude of the theta input (Equation 2, gain $G_{θ}$ ) until theta-nested gamma oscillations were impaired or even abolished (Figure 5). This choice was meant to simulate neurodegeneration in the medial septum, which is known to be affected in Alzheimer’s disease, leading to oscillatory disruptions (Nelson et al., 2014; Hampel et al., 2018; Takeuchi et al., 2021). Another possibility would be that neurodegeneration limits the ability of the septal pacemaker neurons to synchronize, thus producing a weaker collective theta rhythm without affecting the maximum amplitude of individual oscillators. Although we simulated this change in our model by reducing the synchronization parameter, the effects on hippocampal oscillations were less pronounced (Figure 5 - figure supplement 5). Linking these different modeling parameters to experimental biomarkers will be important in future work.

Limitations

Even though we took great care in developing a precise representation of the hippocampal formation, the resulting model remains a simplification that could be further enriched. In particular, we deliberately modeled only a single theta generator, while multiple intra- and extra-hippocampal generators are known to co-exist (Kocsis et al., 1999; Hummos and Nair, 2017). We decided to model septal pacemaker neurons projecting to the EC as the main source of hippocampal theta as reported in multiple experimental studies (Buzsáki, 2002; Buzsáki et al., 2003; Hangya et al., 2009; Colgin, 2013). However, experimental findings and previous models have also proposed that direct septal inputs are not essential for theta generation (Wang, 2002; Colgin, 2013; Mysin et al., 2019), but play an important role in phase synchronization of hippocampal neurons. Furthermore, the model does not account for the connections between the lateral and medial septum and the hippocampus (Takeuchi et al., 2021). These connections include the inhibitory projections from the lateral to the medial septum and the monosynaptic projections from the hippocampal CA3 field to the lateral septum. An experimental study has highlighted the importance of the lateral septum in regulating the hippocampal theta rhythm Bender et al., 2015, an area that has not been included in the model. Specifically, theta-rhythmic optogenetic stimulation of the axonal projections from the lateral septum to the hippocampus was shown to entrain theta oscillations and lead to behavioral changes during exploration in transgenic mice. To account for these discrepancies, our model could be extended by considering more realistic connectivity patterns between the medial/lateral septum and the hippocampal formation, including glutamatergic, cholinergic, and GABAergic reciprocal connections (Müller and Remy, 2018), or by considering multiple sets of oscillators each representing one theta generator.

In terms of neuronal cell types, we also made an important simplification by considering only basket cells as the main class of inhibitory interneuron in the whole hippocampal formation. However, it should be noted that many other types of interneurons exist in the hippocampus and have been modeled in various works with higher computational complexity (e.g. Bezaire et al., 2016; Chatzikalymniou et al., 2021). Among these various interneurons, oriens-lacunosum moleculare (OLM) neurons in the CA1 field have been shown to play a crucial role in synchronizing the activity of pyramidal neurons at gamma frequencies (Tort et al., 2007), and in generating theta-gamma PAC (e.g. Neymotin et al., 2011; Ponzi et al., 2023). Additionally, these cells may contribute to the formation of specific phase relationships within CA1 neuronal populations, through the integration between inputs from the medial septum, the EC, and CA3 (Mysin et al., 2019). Future work is needed to include more diverse cell types and detailed morphologies modeled through multiple compartments.

Another limitation of our model concerns synaptic transmission delays, which have been largely neglected and could affect the phase relationships between the medial septum and different hippocampal subfields. Experimental studies have indeed reported time delays in the population activities of connected anatomical structures (i.e. from EC to DG; Mizuseki et al., 2009), with pyramidal cells in downstream areas like CA3 and CA1 preferentially firing at different phases of theta (Dragoi and Buzsáki, 2006). Propagation effects could also depend on the spatial scale of the model. We also decided to represent only a thin coronal slice of the hippocampal formation, and it remains unclear how an anatomically accurate model of the whole structure would behave in terms of propagation of spontaneous and electrically-induced neuronal activity.

Importantly, we did not consider learning through synaptic plasticity, even though such mechanisms could drastically modify synaptic conduction for the whole network (Borges et al., 2017). Even more interestingly, the inclusion of spike-timing-dependent plasticity would enable the investigation of stimulation protocols aimed at promoting LTP, such as theta-burst stimulation (Larson and Munkácsy, 2015). This aspect would be of uttermost importance to make a link with memory encoding and retrieval processes (Axmacher et al., 2006; Tsanov and Manahan-Vaughan, 2009; Jutras et al., 2013) and with neurostimulation studies for memory improvement (Titiz et al., 2017; Solomon et al., 2021).

From the point of view of neurostimulation, future work is needed to extend the current model to a multi-compartment representation of neurites, since axons are known to be preferentially activated by extracellular electrical stimulation (Rattay et al., 2003). Specifically, multi-compartment cable models have been developed to investigate spike initiation and propagation by modeling the axon as a series of resistance-capacitance circuits following its trajectory (Rattay et al., 2003; Joucla and Yvert, 2012; Ashida and Nogueira, 2018). These models are particularly suited to study the effects of extracellular stimulation, which are non-intuitive and depend as a first approximation on the second spatial derivative of the electrical potential along the cell membrane (Rattay, 1986; Rattay et al., 2003; McIntyre et al., 2004; Rattay et al., 2018). Here, we have modeled electrical stimulation as an intracellular current applied equally across all neurons in the targeted area, which is extremely simplified but enables computational tractability. Future developments will focus on developing an equivalent multicompartment model with realistic axonal trajectories while making sufficient simplifications to allow for realistic computation times.

Finally, we likened conditions of low theta input to pathological states characteristic of oscillopathies such as Alzheimer’s disease, as these conditions disrupted all aspects of theta-gamma oscillations in our model: theta power, gamma power, and theta-gamma PAC (Figure 5). However, it should be noted that changes in theta or gamma power in these pathologies are often unclear, and that the most consistent alteration that has been reported in Alzheimer’s disease is a reduction of theta-gamma PAC (for review, see Kitchigina, 2018). Future work should explore the effects of cellular alterations intrinsic to the hippocampal formation and their impact on theta-gamma oscillations.

Outlook

Overall, this new model of the hippocampal formation represents a methodologically innovative basis to further explore multiple neurostimulation strategies that target hippocampal oscillations. Moreover, the limitations discussed above represent important avenues for future refinements, which will require significant work to overcome the costs in terms of computational tractability (e.g. modeling the whole hippocampus, or using multicompartment models of axonal trajectories and dendritic trees). Ultimately, such model refinements should allow the investigation of the effects of extracellular stimulation using charge-balanced biphasic pulses and multipolar electrode configurations with realistic electrode geometries. Most importantly, we believe that our current work may also serve as an inspiration for future computational models of oscillopathies (not necessarily limited to the hippocampus), which could benefit from interfacing abstract sets of synchronizing oscillators and investigating their interactions with biophysically-realistic neurons.

Parameter	Value
Number of oscillators ( $N$ )	250
Center frequency ( $f_{0}$ )	6 Hz
Standard deviation ( $σ$ )	0.5 HZ
Synchronization ratio ( $k / N$ )	15
Phase reset gain ( $G_{r e s e t}$ )	4
Peak phase ( $θ_{p e a k}$ )	0 rad
Firing rate time constant ( $τ_{F R}$ )	10 ms

Parameter	Expression
$A$	${29.10}^{3} μ m^{2}$
$C_{m}$	$1 μ F / {c m}^{2}$
$g_{L}$	$0.01 {m S / c m}^{2}$
$E_{L}$	$- 70 m V$
$g_{K}$	$5 {m S / c m}^{2}$
$E_{K}$	$- 100 m V$
$α_{n, K}$	$- 0.032 \frac{V_{m} + 40 m V}{- 1 + e^{- 0.02 (V_{m} + 40 m V)}}$
$β_{n, K}$	$0.5 e^{\frac{- (V_{m} + 45 m V)}{40 m V}}$
$g_{N a}$	$50 {m S / c m}^{2}$
$E_{N a}$	$50 m V$
$α_{m, N a}$	$- 0.32 \frac{V_{m} + 42 m V}{e^{- \frac{V_{m} + 42 m V}{4 m V}} - 1}$
$β_{m, N a}$	$0.28 \frac{V_{m} + 15 m V}{e^{- \frac{V_{m} + 15 m V}{5 m V}} - 1}$
$α_{h, N a}$	$0.128 e^{- \frac{V_{m} + 38 m V}{18 m V}}$
$β_{h, N a}$	$\frac{4}{1 + e^{- \frac{V_{m} + 15 m V}{5 m V}}}$
$g_{M}$	$90 μ {S / c m}^{2}$
$E_{M}$	$- 100 m V$
$p_{\infty}$	$\frac{1}{1 + e^{- 0.01 (V_{m} + 35 m V)}}$
$τ_{p}$	$\frac{1}{3.3 e^{\frac{(V_{m} + 35 m V)}{20 m V}} + e^{- \frac{(V_{m} + 35 m V)}{20 m V}}}$
$g_{C a}$	$0.1 {m S / c m}^{2}$
$E_{C a}$	$120 m V$
$α_{m, C a}$	$- 0.055 \frac{V_{m} + 27 m V}{e^{- \frac{V_{m} + 27 m V}{17 m V}} - 1}$
$β_{m, C a}$	$- 0.94 e^{\frac{V_{m} + 75 m V}{17 m V}}$
$α_{h, C a}$	$- 0.000457 e^{\frac{V_{m} + 13 m V}{50 m V}}$
$β_{h, C a}$	$\frac{0.0065}{e^{- \frac{V_{m} + 15 m V}{28 m V}}}$
$g_{C A N}$	$25 μ {S / c m}^{2}$
$E_{C A N}$	$- 20 m V$
$α_{m, C A N}$	$0.0002 e^{1.4 \frac{[C a]_{i}^{2 +}}{0.5 m o l / L}}$
$β_{m, C A N}$	$0.0002 e^{1.4}$

Parameter	Expression
$A$	${14.10}^{3} μ m^{2}$
$C_{m}$	$1 μ {F / c m}^{2}$
$g_{L}$	$0.1 {m S / c m}^{2}$
$E_{L}$	$- 65 m V$
$g_{K}$	$9 {m S / c m}^{2}$
$E_{K}$	$- 90 m V$
$α_{n, K}$	$0.01 \frac{V_{m} + 34 m V}{1 - e^{- 0.1 (V_{m} + 34 m V)}}$
$β_{n, K}$	$0.125 e^{- \frac{V_{m} + 44 m V}{80 m V}}$
$g_{N a}$	$35 {m S / c m}^{2}$
$E_{N a}$	$55 m V$
$α_{m, N a}$	$0.1 \frac{V_{m} + 35 m V}{1 - e^{- 0.1 (V_{m} + 35 m V)}}$
$β_{m, N a}$	$4 e^{- \frac{V_{m} + 60 m V}{18 m V}}$
$α_{h, N a}$	$0.07 e^{- \frac{V_{m} + 58 m V}{20 m V}}$
$β_{h, N a}$	$\frac{1}{1 + e^{- 0.1 (V_{m} + 28 m V)}}$

	E→E	E→I	I→E	I→I
EC		20 pS	600 pS
DG		180 pS	1800 pS
CA3	20 pS	20 pS	600 pS
CA1		60 pS	1800 pS	1800 pS

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Dynamical computational model of the hippocampal formation and medial septum oscillatory drive.

Emergence of gamma oscillations in coupled excitatory-inhibitory populations under ramping input to both populations.

Theta-nested gamma oscillations and theta phase reset in response to a single stimulation pulse.

CAN and adaptation (M) currents modulate neuronal responses to single-pulse stimulation in the absence of theta input.

Medial septum oscillatory drive and stimulation amplitude govern the steady-state response to single-pulse stimulation.

Single-pulse stimulation phase differentially affects network responses depending on the presence of theta phase reset.

Pulse train stimulation restores theta-nested gamma oscillations depending on stimulation timing and theta phase reset.

Full list of the default parameter values for the Kuramoto oscillators.

Full list of parameter values and expressions for pyramidal neurons.

Full list of parameter values and expressions for interneurons.

A pre-synaptic spike causes an increase in the conductances he and hi in the post-synaptic neuron.

A pre-synaptic spike causes an increase in the conductances he and hi in the post-synaptic neuron.

Number of neurons per subfield of the hippocampal formation, divided by neuron type.

Maximum probability of connection (Aintra, Equation 14) between neurons within each region.

Inter-area connection strengths (Ainter).

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Nikolaos Vardalakis

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Amélie Aussel

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Nicolas P Rougier

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Fabien B Wagner

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A pre-synaptic spike causes an increase in the conductances $h_{e}$ and $h_{i}$ in the post-synaptic neuron.

A pre-synaptic spike causes an increase in the conductances $h_{e}$ and $h_{i}$ in the post-synaptic neuron.

Maximum probability of connection ( $A_{i n t r a}$ , Equation 14) between neurons within each region.

Inter-area connection strengths ( $A_{i n t e r}$ ).