Simulation of oscillatory dynamics induced by an approximation of grid cell output

Overall model structure

The program used for simulations in this paper was adapted from the program used in a model of the lateral entorhinal cortex (LEC) (Traub and Whittington 2022). The reader is referred to that paper and an earlier one (Traub et al. 2021) for an account of the overall approach. The major modifications in the present program, in contrast to LEC were these: (1) conversion of model fan cells to model stellate cells, these latter endowed with intrinsic membrane properties that permit intrinsic theta oscillations over some potential range (Alonso and Klink 1993; Alonso and Llinás 1989; Dickson et al. 2000b; Fransén et al. 2004; Haas and White 2002; Klink and Alonso 1997; Magistretti and Alonso 1999; van der Linden and Lopes da Silva 1998); (2) alterations of synaptic connectivity and synaptic conductances, in an attempt to harmonize with recent data in MEC based on multiple pair recordings (reviewed in Tukker et al. (2022)); (3) the use of highly structured driving currents to principal neurons, in contrast to stochastic afferent pulses, so as to replicate grid cell intracellular recordings in head-fixed mice in virtual environments (Domnisoru et al. 2013; Schmidt-Hieber & Häusser 2013a, b); (4) periodic driving currents to certain interneurons (in some cases also principal neurons), to emulate inputs from the septum (Gonzalez-Sulser et al. 2014). Details concerning these modifications are spelled out below.

Modeling principles, briefly reiterated

The model incorporates structure and dynamics at membrane, cellular, and network levels. There are a number of neuronal types, each with multiple (dozens) of compartments that are axonal, somatic, or dendritic. The number and interconnection patterns of the compartments are, in general, distinct for each neuronal type. Every compartment contains passive and active membrane conductances, the latter with voltage- (and sometimes also Ca²⁺)-dependent kinetics, following Hodgkin–Huxley-like descriptions. Model neurons interact with each other through simulated chemical synapses, using glutamate (AMPA and NMDA) or GABA (GABA_A and GABA_B) receptors, with connections forming at cellular compartments that are determined by the combination of presynaptic and postsynaptic cell types. Model neurons may also interact via non-rectifying electrical coupling that simulates gap junctions. These latter interactions occur in dendrite/dendrite for homologous interneuron types, and axon/axon for homologous principal neuron types (Traub et al. 2005, 2018). It is important to emphasize that all synaptic connectivity is random – we are not attempting to produce grid firing patterns through constrained network structure.

Neuron types and numbers

The model network contained 1000 layer 2 pyramidal neurons (L2pyr), 500 stellate cells, 500 layer 3 pyramidal neurons (L3pyr), 200 superficial interneurons (100 “VIP” and 100 neurogliaform), 200 deep fast-spiking basket cells, 100 deep “LTS” (low-threshold spiking, dendrite-contacting) interneurons and 200 deep neurogliaform cells. We did not include chandelier cells (Soriano et al. 1993) in the present implementation. Unlike the lateral entorhinal cortex model (Traub and Whittington 2022), we did not use stochastic stimulation of afferents. Except for stellate cells, the intrinsic properties of the model neurons remained as before, unless noted otherwise. In particular, the repertoire of membrane conductances, passive membrane parameters and ionic conductance details were as before (other than stellate cells – see below). This was likewise the case for the schemas used to simulate synaptic conductances. Tables for representative synaptic conductance scaling factors and connectivity values, for one particular simulation (MEC41.f) are given in Tables 1 and 2; these are analogous to Tables 1 and 2 of our lateral entorhinal cortex paper (Traub and Whittington 2022).

Stellate cell specifics

As for fan cells in a previous model of lateral entorhinal cortex (Traub and Whittington 2022), the model used here for stellate cells ultimately derives from that used for superficial regular spiking pyramidal cells (Traub et al. 2005), with basal dendrites disconnected. Stellate cells had a GABA_A reversal potential of −55 mV, as in the case of model fan cells (Nilssen et al. 2018; Traub and Whittington 2022). The integration subroutine in the code is called integrate.MECstell.f. Because of intrinsic theta oscillations in stellate cells, we paid particular attention to the two most relevant conductances (Dickson et al. 2000a), persistent Na⁺ (g_Na(P)) and the h conductance (called “g_AR” in the code, for “anomalous rectifier”). g_Na(P) does not inactivate, has activation kinetics as for transient g_Na, and densities (in mS/cm²) of 0.3 on the soma, 0.25 on the proximal apical shaft, and 0.1875 on the next-proximal apical shaft and oblique dendrites. This density is larger than that used by Fransén et al. (2004), ∼0.04 mS/cm². In computing g_Na(P) conductance, the activation variable “m” is raised to the 1st power, in contrast to 3rd power for the transient Na⁺ conductance. The h conductance does not inactivate and follows Hodgkin–Huxley-like kinetics with the activation variable “m” raised to the first power, and reversal potential −35 mV. The kinetics of h conductance are illustrated in Figure 1. Its density is 0.25 mS/cm² on the soma and dendrites, slightly larger than the ∼0.15 mS/cm² of Fransén et al. (2004). Another notable difference from the model of Fransén et al. (2004) was our use of a larger g_K(AHP) density (0.6 vs. 0.01 mS/cm²). The input resistance of the present model stellate neuron, with all conductances active, and estimated by passage of steady depolarizing currents into the soma, was 15 MΩ, smaller than in the model of Fransén et al. (2004) and probably due to our use of a larger g_K(AHP) density.

Figure 1:

Properties of model stellate neuron. (A) Somatic potential of single model stellate cell in response to current injections (red: 0.8 nA, black 0.71 nA). Note that the cell can generate intrinsic theta (∼6 Hz in this case) that can be either sub- or suprathreshold for action potentials, but the behavior is parameter-sensitive. (B) Kinetic properties of model h current, h_∞ (steady-state value of activation variable) and τ_h (time constant). Only one component for the kinetics was used, unlike Dickson et al. (2000b). Simulation MECprelim37.

Detailed kinetics for stellate g_Na(P) were these: m_∞ = 1/(1 + exp (–V–38)/10); and

τ_m = 0.025 + 0.14 exp ((V + 30)/10) if V < −30 mV, 0.02 + 0.145 exp((–V–30)/10) otherwise.

For the h-conductance: m_∞ = 1/(1 + exp((V + 75)/5.5)) and

τ_m = 0.25/(exp (−14.6–0.086 V) + exp(−1.87 + 0.07 V)) [V in mV, τ in ms].

An example of firing behavior of a single isolated model stellate neuron, for 2 steady somatic driving currents, is shown in Figure 1. We found that intrinsic theta was sensitive to the driving current in this model. Subthreshold oscillations appear in the model stellate neuron at potentials of about −57 mV; this compares well to Alonso and Klink (1993) at “positive to −60 mV” and Dickson et al. (2000b) at “positive to about −55 mV”. We did not examine noise-driven aspects of the intrinsic oscillations, unlike Fransén et al. (2004).

Driving currents

Many network simulations included a theta-frequency sinusoidal drive (period 120 ms, 8.33 Hz) to basket and LTS interneurons (and sometimes just to principal cells, or both principal cells and interneurons). This drive was in the form of a somatic current, typically −0.15 nA maximum amplitude, and not phase-shifted across the respective interneuron populations. (The absence of phase shift stands in contrast to Hasselmo and Shay (2014) – see their Figure 3.) It was intended to represent input from the septal nuclei. Grid cell activity was simulated with a driving current to subpopulations (half of them) of the principal cells (the shape is shown in Figure 4). It took the form of a depolarizing rectified cosine somatic current, period 1 s, peak amplitude 1.0–1.5 nA. This time-varying drive was superimposed on a tonic hyperpolarization in some simulations, in order to prevent excessive firing (up to −0.3 nA). The drive to individual cells was phase-shifted so as to cover one cycle across each respective subpopulation of principal neurons.

In some simulations “ectopic spikes” – spontaneous axonal action potentials – were included as a noise source; these were also simulated with Poisson-distributed brief axonal current pulses, as in previous publications (Traub et al. 1995, 1999a). Small tonic bias currents, randomly distributed, were likewise delivered to most of the neurons, in order to introduce heterogeneity into the system.

Software applications

Power spectra for analysis of oscillations were computed with the Fortran package Dfftpack, a double precision version of fftpack, as supported by IBM. The simulation data base consisted of 87 single-cell and network simulations. Code was written in Fortran and is available from rtraub@us.ibm.com. The main programs were called MECprelim.f and MEC.f. Programs run in the mpi parallel environment. Differential equations were integrated with a 2nd order Taylor series method with time step 2 µs. Programs ran on an IBM Power GPU compute node residing in the IBM Cognitve Computing Cluster (CCC), under the Linux operating system. A simulation of 2.5 s of network activity ran for approximately 5.52 h. [Code has been deposited at the Yale NEURON website, senselab.med.yale.edu (ModelDB), accession #267585. The files there include the makefile and run command, as well as a sample pdf of output.]

Experimental methods

Previously unpublished experimental data are illustrated below, taken from an experimental model of spontaneous in vitro slow oscillations in superficial entorhinal cortex. These data were obtained during the course of the study published in Cunningham et al. (2006), and detailed methods are described in that paper.

Slow depolarizations with superimposed fast oscillations are a widespread phenomenon in the mammalian brain

Before describing the detailed behaviors of our model, we wish to establish a larger context for grid cell activities – not the functional aspects of these activities, that is their relationship to a spatial environment and navigation, but rather the purely electrophysiological aspects. We touch on this larger context here in order to provide justification for our electrophysiologically oriented approach. The main idea is that there are numerous examples in the mammalian brain of slow, ramp-like depolarizations in principal neurons, associated with fast oscillations; such slow potentials are sometimes rhythmic on a time scale of seconds, as is the case with grid cells. Some examples are illustrated in Figure 2. Thus, Figure 2A is an intracellular recording (from Jagadeesh et al. 1992) from a visual cortical neuron in an anesthetized cat, as a moving bar of correct orientation is swept back and forth (action potentials are truncated). The original paper demonstrates that gamma field potential oscillations are present during the depolarizations. The lower traces in Figure 2B are intracellular recordings from a hippocampal place cell (head-fixed behaving mouse, Bittner et al. 2017), with the averaged ramp potential shown on the right. [Note that place cell activity in rodents is associated with field oscillations at theta and gamma frequencies (Ahmed and Mehta 2012; Csicsvari et al. 2003).] Figure 2C shows extracellular (ec) and simultaneous intracellular L3pyr recordings from the in vitro medial entorhinal cortex, bathed in a blocker of AMPA receptors (Cunningham et al. 2006). A slow oscillation is apparent (note the 5 s time scale). In the original paper, it is shown that oscillatory field activity up to ∼35 Hz occurs during the Up states (Figure 2 of Cunningham et al. 2006; see also Figure 5 below), and that this particular slow oscillation is dependent upon kainate receptors. Figure 2D shows the intracellular correlates of a tetanically induced CA1 hippocampal slice gamma oscillation, in an interneuron (“i-cell”) and a pyramidal neuron (“e-cell”) (Whittington et al. 1997a, b); in this case, the slow depolarization is dependent upon metabotropic glutamate receptors and M1 cholinergic receptors. Because the slow depolarizations in Up states employ various receptor types, we chose to be agnostic on the synaptic source(s) of excitatory drives to grid cells, and instead used simply time-varying injected currents in our network model.

Figure 2:

Examples of slow depolarizations in cortical structures. (A) Intracellular potential (spikes truncated) of visual cortical neuron during visual stimulation (complex cell, cat area 17; bright bar moving at preferred orientation). (B) Intracellular recordings from CA1 hippocampal neuron, head-fixed mouse running on a virtual linear track. On the left, 3 successive traces show potential on a lap before a plateau, the plateau itself, and place-cell firing on a subsequent lap. The trace on the right is an average of potentials that shows the ramp depolarization. (C) Extracellular (ec) and intracellular (ic) potentials from layer 3 entorhinal cortical neuron (rat in vitro), showing spontaneous slow oscillation. The recording was obtained in the presence of the AMPA receptor blocker SYM 2206. The depolarizing phase requires kainate-type glutamate receptors and ATP-gated K⁺ currents contribute to the hyperpolarizing phases. (D) Intracellular slow depolarizing potentials from interneuron (i-cell) and pyramidal cell (e-cell), rat CA1 hippocampus in vitro, in response to local tetanic stimulation (stimulus artifacts at beginning of the traces). The slow depolarizations in this case were mostly due to activation of metabotropic glutamate receptors. Reproduced with permission from Jagadeesh et al. (1992) (A); Bittner et al. (2017) (B); Cunningham et al. (2006, Copyright National Academy of Sciences) (C); Whittington et al. (1997a, b) (D).

Preliminary tests of the model: synchronized burst and persistent gamma oscillations

The network model was tested through its ability to replicate two types of collective behavior: synchronized bursts in layer 3 and gamma oscillations. Figure 3A shows a synchronized burst induced by a brief, large current pulse to 100 model layer 3 pyramidal neurons (L3pyr), when GABA_A conductances are blocked in principal model neurons. This can be compared with recordings obtained in bicuculline by Gloveli et al. (1997). There is, however, an important caveat: the simulation in Figure 3A was run without electrical coupling between L3pyr cells, while such coupling is observed experimentally in layer 3 (Dhillon and Jones 2000) – even possibly strong enough to allow spike transmission from one neuron to another. When the simulation of Figure 3A was run with L3pyr electrical coupling present (not shown) using 8 nS conductance and having each axon coupled to 2 others on average, prolonged synchronized bursts occurred with runs of spikelets. This was unlike the illustrations of Gloveli et al. (1997). One possible interpretation is that L3pyr electrical coupling, while present, may be below the percolation threshold of connectivity, that is each cell couples to less than one other cell on average (Traub et al. 1999a).

Figure 3:

Tests of the model. (A) Synchronized burst during block of GABA_A synaptic inputs to all model principal cells. A 10 ms current pulse was delivered to 100 L3pyr (bottom trace). The black intracellular potential was from a stimulated L3pyr cell; the red one was a recruited cell. Middle trace shows the L3pyr “field” (inverted average of all L3pyr somatic voltages). Gap junctions were not present in L3pyr but were for L2pyr and stellate cells (8 nS conductance, each cell couples to 2 others on average). Compare Figures 5 and 7 of Gloveli et al. (1997). (B) Simulated gamma oscillation with ectopic spikes and axonal gap junctions, c.f. Cunningham et al. (2003, 2004 and Figure 10 below. Parameters: each axon (L2pyr, stellate, L3pyr) couples to 2 other homologous axons, on average, with conductance 8 nS; ectopic axonal spikes at average 2 Hz for L2pyr, 1 Hz for stellates and L3pyr. (C) Power spectra of L2pyr “fields” for simulation in B (black trace) and repeated simulation with I_h blocked in stellate cells (red trace). Note the reduction in theta after such block, consistent with experiment. Simulations MEC17, MEC24, MEC25.

Figure 3B, C illustrates a simulated gamma oscillation, with a theta component, when ectopic spikes (see Methods) were present, as well as open gap junctions (conductance 8 nS) in principal cells (above the percolation limit, i.e. each cell couples to 2 other homologous cells, on average). Spikelets occur, as observed by Cunningham et al. (2004), and additionally, the theta frequency component attenuates when I_h is blocked in stellate cells (Figure 3C) – also consistent with experiment (c.f. Cunningham et al. 2003).

Simulated grid cell activity with imposed slow rectified sinusoids and theta-frequency inputs to deep interneurons.

Fast (∼30 Hz) subthreshold potentials, spikes on the Up states, emergence of a ∼20 Hz network oscillation as well. Once the basic features of the network model were tested, we examined the effects of combined external drives to the system. As described in more detail in Methods, there were two external drives: (1) a slow (period 1 s) oscillating drive to one half of each of the principal cell populations (L2pyr, stellate, L3pyr), with the drive to individual cells phase-shifted to cover one cycle across each of these subpopulations – this to simulate grid cells; (2) periodic theta-frequency currents (not phase-shifted) to the deep basket cells and LTS interneurons. The resulting potentials in a selected L2pyr and stellate cell are shown in Figure 4A – the potentials closely resemble experimental intracellular recordings of entorhinal grid cells in 1D environments (Schmidt-Hieber and Häusser 2013a, b; Domnisoru et al. 2013), containing as they do action potentials (truncated in the figure) on the depolarizing phases, as well as high-frequency synaptic potentials. Figure 4B shows the power spectra of averaged L2pyr and stellate activities. Theta is most prominent in the stellate population, while a ∼30 Hz oscillation occurs in both subpopulations. There are as well peaks at ∼20 Hz which, as seen below (Figure 5), have also been seen experimentally in vitro. In our model, when the theta-frequency input is eliminated, the ∼20 Hz peak disappears (in addition, of course, to the theta peak) (not shown). The ∼20 Hz peak therefore presumably results from an interaction between two oscillators, theta and gamma/beta. It does not appear to arise via period concatenation (c.f. Roopun et al. 2008).

Figure 4:

Simulated grid firing behavior. Slow periodic (1 Hz) depolarizing currents were delivered to half the L2pyr, half the stellates, and half the L3pyr. One such drive (not for the cells shown) is illustrated. The phase of the driving currents was shifted for each cell so that a full cycle was covered across each principal cell population (see figures below). Theta-frequency currents were also delivered to basket cells and LTS interneurons (see Methods) to simulate septal input; these currents were not phase shifted between neurons. Gap junctions in principal cells were blocked. (A) Intracellular potentials from an L2pyr and a stellate cell (action potentials truncated) – they closely resemble experimental recordings of Schmidt-Hieber and Häusser (2013a, b) and Domnisoru et al. (2013). (B) Power spectra of the L2pyr (black) and stellate (red) average somatic potentials. The β/γ peaks at ∼30 Hz are inhibition-based (see Figure 8). The peaks at ∼20 Hz do not appear to be inhibition-based and may result from intrinsic currents. Simulation MEC41.

Figure 5:

Experimental slow oscillation in the entorhinal cortex is accompanied by fast network oscillations during the Up state. (A) Dual extracellular local field potential and intracellular recordings in the superficial layers of the entorhinal cortex reveal a spontaneous slow oscillation during which layer 3 pyramidal cells alternate between hyperpolarized (Down state) and depolarized (Up state). The pyramidal cell Up states are coincident with the active phase of the network slow oscillation which has contained within it fast network activity. (B) Band pass filtering of fast oscillatory components within the slow oscillation (dashed red box) reveals temporally organized oscillations that are associated with the Up state. Fast Fourier analysis reveals, as shown in the power spectrum below, activity organized in the theta, beta, and gamma domains as predicted by the computational modelling studies (c.f. Figure 4).

Fast oscillations associated with a spontaneous slow oscillation in the in vitro rat superficial entorhinal cortex.

We have previously shown (Figure 3A of Cunningham et al. 2006) that oscillatory activity occurs during the Up states of the spontaneous in vitro entorhinal cortex oscillation. Figure 5 provides further detail of this experimental finding, in the form of a power spectrum of the oscillatory activity. It is interesting that multiple peaks occur in the experimental power spectrum, with values similar to those we observed in the computational model (Figure 4B).

Overview of simulated population activity during grid cell drives (∼1 cycle) to half the principal cells.

The population of “grid cells” encodes position but interneurons do not. Theta frequency pauses are apparent, as is a synchronized fast oscillation. Figure 6 gives a portrait of the spatiotemporal structure that emerges in the network model over one “grid” cycle (the red box). The phase shifts among the model grid cells are apparent, so that the population activity encodes time (and hence space). Several other features are apparent. Due to the theta frequency drive to interneurons, the interneuron spikes are grouped between pauses, which in turn give rise to pauses in the “grid cell” action potentials (horizontal purple arrow). Additionally, the ∼30 Hz time scale is apparent in the stellate raster (blue rectangle). Finally, the action potentials of the basket cells are highly synchronized, so that phase shifts between basket cells are negligible. Any temporal structure that exists amongst these interneurons is the result of the external theta frequency drive and is not related to drives to the “grid cells”.

Figure 6:

Imposed and emergent spatiotemporal structure in the grid cell network model. The figure shows raster plots of spike times from the simulation of Figure 4: Some of the “grid” stellate cells (i.e. those receiving external 1 Hz periodic drives at various phases), some of the “not-grid” cells (not receiving periodic drives), and some of the basket cells. The network shows structure on different time scales: (1) on the scale of grid periodicity (red box), with the phase shifts apparent; (2) on a theta frequency scale (horizontal purple arrow); (3) on a β/γ scale (blue box). Of additional note: the network induces synchrony of the not-grid cells, and also of the basket cells. In particular, there are no systematic phase shifts in basket cell firings, although there are pauses involving the whole basket cell population (see also Figure 7).

Theta-modulated β/γ oscillation in deep interneurons during simulated grid cell activity

The grouping of interneuron spikes by the imposed theta oscillation is shown in more detail in Figure 7A. This figure additionally illustrates the detailed appearance of synaptic potentials in an L2pyr “grid cell”, as well as the synaptic conductances “seen” by one of the model basket cells. For comparison, Figure 7B illustrates synaptic potentials in a hippocampal fast-spiking interneuron during a network gamma oscillation that was induced by tetanic stimulation. (This experimental model of gamma oscillations was introduced in Traub et al. (1996a) and further developed in Whittington et al. (1997a, b, 2001). See also Figure 2D. For review, please see the monograph by Traub et al. 1999b.) At least qualitatively, there is agreement between the present network model and the tetanic experimental model. We note that the prominent phasic EPSCs in the model interneurons argue against the type of gamma generated by interneuron networks without such EPSCs (Whittington et al. 1995; Traub et al. 1996b), so-called “ING” or interneuron network gamma. Instead, the simulated β/γ oscillations appear to depend on pyramid cell inputs.

Figure 7:

Oscillatory interneuron behavior during simulated grid cell activity and during experimental tetanic gamma oscillation. (A) Data from the simulations of Figures 4 and 6. Theta-modulated β/γ activity amongst basket cells results from the external (septal) drive to the interneurons, combined with the ability of the principal cell/interneuron populations to generate network oscillations. (B) Synaptic potentials in a stratum pyramidale fast-spiking interneuron during experimental tetanic gamma oscillation, from (Whittington et al. 2001), with permission. Compare with the synaptic conductances developed in a model fast-spiking (basket) interneuron in (A).

We should note that our model involves mechanisms that are distinct from the experimental conditions used by Pastoll et al. (2013). The latter authors optogenetically stimulated stellate cells and fast-spiking interneurons of MEC in vitro, but not pyramidal cells. In contrast, our model uses strong stimulation of pyramidal cells as suggested by intracellular grid cell recordings. The fast oscillations observed by Pastoll et al. (2013), about 85 Hz are considerably faster than the ∼30 Hz oscillations in our model, but it is hard to make a mechanistic comparison because of the different conditions.

In the grid cell network model, fast network oscillations virtually disappear when inhibition from deep interneurons to principal cells is blocked.

Most, if not all, experimental models of gamma oscillations are “inhibition-based” (Whittington et al. 2000), disappearing upon blockade of GABA_A receptors. As Figure 8 illustrates, the present ∼30 Hz oscillation generated in our grid cell simulations is also inhibition-based – it disappears with blockade of GABA_A conductances in principal cells. Interestingly, the simulation shown in Figure 8Aii, in disinhibited conditions, did not exhibit synchronized bursts; this was presumably a consequence of the limited recurrent synaptic excitation in our model, along with the lack of a strong synchronized afferent stimulus (unlike what is seen in Figure 2A).

Figure 8:

The γ/β oscillations in the model grid cell network are inhibition-based. (Ai) data from the simulation of Figures 4, 6 and 7 showing rhythmic IPSCs in a model stellate cell. (Aii) Data from an additional simulation identical to that of (Ai), except for blockade of GABA_A receptor-mediated inhibition from deep interneurons (basket cells, LTS interneurons, deep neurogliaform cells) to principal neurons. (B) Power spectra of the stellate “fields” for the simulations of (Ai and Aii). Blocking IPSCs from deep interneurons virtually abolishes the γ/β oscillations.

Similarity of simulated grid cell activity in stellate neuron to in vitro stellate activity during spontaneous slow oscillation.

It is striking that, at least qualitatively, there is a similarity between the intracellular activities of in vivo grid cells (Domnisoru et al. 2013; Schmidt-Hieber and Häusser 2013a), simulated grid cells (e.g. Figure 8Ai), and intracellular stellate cell recordings during spontaneous slow MEC oscillations in vitro. This is further illustrated in Figure 9, which shows (with in vitro stellate cell recordings) slow repeating depolarizations, having superimposed action potentials and apparent synaptic potentials. Such a similarity suggests that in vivo grid cell activity may engage cellular mechanisms that are common to the in vitro slow oscillation model (Cunningham et al. 2006). A possible similarity would be the use of kainate receptors as a contributor to the slow depolarization, a mechanism shown to be operative in vitro (Cunningham et al. 2006).

Figure 9:

Experimental entorhinal cortex stellate cells behave in a bistable manner during in vitro slow oscillations. (A) Intracellular recordings in layer 2 entorhinal cortex stellate cells reveal that they demonstrate periods of silence and activity; data from the spontaneous in vitro MEC activity illustrated in Figure 5 (Cunningham et al. 2006). The periods of activity correspond to the slow oscillation and accompanying fast network oscillations. (B) Intracellular traces show examples of activity from three different stellate cell recordings. During the in vitro slow oscillation, stellate cells generate action potentials, increased membrane oscillations and increased synaptic activity. There is a striking qualitative similarity to simulated grid cell activity (c.f. Figure 8Ai).

When the grid cell network model includes electrical coupling between principal cell axons, spikelets become especially prominent.

There is direct (Dhillon and Jones 2000) and indirect (Cunningham et al. 2004) evidence for electrical coupling between principal entorhinal cortical neurons, particularly in layer 3. For this reason, we repeated the simulation illustrated in Figure 4 – which had been run without principal cell gap junctions – so that gap junctions were present (Figure 10). Comparing the upper and middle traces of Figure 10, it is apparent that spikelets become apparent with the opening of axonal gap junctions. Indeed, this is not surprising (compare, e.g., Cunningham et al. (2004)). Further blocking of GABA_A receptors (bottom trace of Figure 10) then leads to high-frequency flurries of spikelets. It is interesting that in our simulation conditions for Figure 10, blocking GABA_A receptors did not lead to synchronized epileptiform bursts (recall that in Figure 3, a large current pulse was delivered to a subpopulation of L3pyr, unlike the case in Figure 10). This is presumably the result of the limited recurrent excitatory chemical synaptic connectivity used in the present model.

Figure 10:

In simulations, principal cell axonal gap junctions increase the incidence of spikelets. Upper trace: Data from the simulation of Figure 4. Middle trace: Data from a repeated simulation with principal cell axonal gap junctions. Each L2pyr, stellate cell, and L3pyr couples to an average of 2 other homologous cells, with conductance 8 nS. Ectopic spike rates per axon were, respectively, 2 Hz, 1 Hz, and 1 Hz. Note the emergence of spikelets (e.g. asterisk). Bottom trace: The middle simulation was repeated, now with blockade of GABA_A inputs to principal cells from deep interneurons. Now bursts of spikelets occur, but not typical synchronized bursts – a likely consequence of the limited recurrent synaptic excitation. Simulations MEC41, MEC43, and MEC44.

Experimental evidence for spikelets in the spontaneous slow oscillation MEC model

It is interesting that spikelets do occur in entorhinal cortex stellate cells during the spontaneous (drug-free) slow oscillation of the sort illustrated in Figure 5 and described in Cunningham et al. (2006) (Figure 11). These spikelets tend to occur prior to the Up states, at a time when kainate receptors appear to be activated. How the putative electrical coupling underlying the spikelets might contribute to the Up state, and likewise how kainate receptors might contribute to grid-associated depolarizations, are matters that remain to be investigated.

Figure 11:

Experimental evidence for electrical coupling between medial entorhinal cortex stellate cells preceding the transition to the Up state. (A) Dual extracellular local field potential (above) and intracellular recording (below, action potential truncated) in the superficial layers of the entorhinal cortex demonstrate the presence of spikelets in a layer 2 stellate cell during the transition from the Down state to the Up state, clearly visible with expanded time scale below. The blue box identifies the Up state and the red box identifies activity leading into the Up state. (B) These spikelets (red dots) manifest as brief (c. 1 ms) and small (c. 0.5 mV) depolarizations that occur in the period preceding the neuron’s active phase which is correlated with the extracellular slow oscillation.

Gamma oscillations in the entorhinal cortex

Gamma oscillations have been reported in layer 2 of MEC during spatial behavior wherein grid cell firing would be expected to occur (Chrobak and Buzsáki 1998; Figures S7C and S8A of Fernández-Ruiz et al. 2021), although both of these studies focused on oscillations (89 Hz and faster) that were of higher frequency than the ∼30 Hz ones exhibited in our model. To the best of our knowledge, there have been no explicit correlations of gamma oscillations with grid cell firings. Medial entorhinal gamma oscillations are also known to occur in vitro, in the presence of kainate (Cunningham et al. 2003, 2004; Klein et al. 2016), and of carbachol in the isolated guinea pig brain (Dickson et al. 2000a), as well as in association with spontaneous slow oscillations (Cunningham et al. 2006, and illustrated above, Figures 5, 9 and 10). It would be useful to have recordings of MEC gamma oscillations that are specifically correlated with grid cell activities.