An Analysis of Waves Underlying Grid Cell Firing in the Medial Enthorinal Cortex
- Mayte Bonilla-Quintana^{1}Email authorView ORCID ID profile,
- Kyle C. A. Wedgwood^{2}View ORCID ID profile,
- Reuben D. O’Dea^{1}View ORCID ID profile and
- Stephen Coombes^{1}View ORCID ID profile
https://doi.org/10.1186/s13408-017-0051-7
© The Author(s) 2017
Received: 9 March 2017
Accepted: 8 August 2017
Published: 25 August 2017
Abstract
Layer II stellate cells in the medial enthorinal cortex (MEC) express hyperpolarisation-activated cyclic-nucleotide-gated (HCN) channels that allow for rebound spiking via an \(I_{\text{h}}\) current in response to hyperpolarising synaptic input. A computational modelling study by Hasselmo (Philos. Trans. R. Soc. Lond. B, Biol. Sci. 369:20120523, 2013) showed that an inhibitory network of such cells can support periodic travelling waves with a period that is controlled by the dynamics of the \(I_{\text{h}}\) current. Hasselmo has suggested that these waves can underlie the generation of grid cells, and that the known difference in \(I_{\text{h}}\) resonance frequency along the dorsal to ventral axis can explain the observed size and spacing between grid cell firing fields. Here we develop a biophysical spiking model within a framework that allows for analytical tractability. We combine the simplicity of integrate-and-fire neurons with a piecewise linear caricature of the gating dynamics for HCN channels to develop a spiking neural field model of MEC. Using techniques primarily drawn from the field of nonsmooth dynamical systems we show how to construct periodic travelling waves, and in particular the dispersion curve that determines how wave speed varies as a function of period. This exhibits a wide range of long wavelength solutions, reinforcing the idea that rebound spiking is a candidate mechanism for generating grid cell firing patterns. Importantly we develop a wave stability analysis to show how the maximum allowed period is controlled by the dynamical properties of the \(I_{\text{h}}\) current. Our theoretical work is validated by numerical simulations of the spiking model in both one and two dimensions.
Keywords
1 Introduction
The ability to remember specific events occurring at specific places and times plays a major role in our everyday life. The question of how such memories are established remains an active area of research, but several key facts are now known. In particular, the 2004 discovery of grid cells in the medial enthorinal cortex (MEC) by Fyhn et al. [2], supports the notion of a cognitive map for navigation. This is a mental representation whereby individuals can acquire, code, store, recall, and decode information about relative spatial locations in their environment. The concept was introduced by Tolman in 1948 [3], with the first neural correlate being identified as the place cell system in the hippocampus [4]. Place cells are found in the hippocampus and fire selectively to spatial locations, thereby forming a place field whose properties change from one environment to another. More recently, a second class of cells was identified that fire at the nodes of a hexagonal lattice tiling the surface of the environment covered by the animal—these are termed grid cells [5]. As an animal approaches the centre of a grid cell firing field, the spiking output of grid cell will increase in frequency. The grid field size and spacing increases from dorsal to ventral positions along the MEC and is independent of the animal’s speed and direction (even in the absence of visual input) and independent of the arena size. In rats, the grid field spacing can range from roughly 30 cm up to several meters [6]. Other grid cell properties include firing field patterns that manifest instantly in novel environments and maintain alignment with visual landmarks. Furthermore, neighbouring grid cells have firing fields with different spatial phases, whilst grid cells with a common spacing also have a common orientation (overturning an original suggestion that they have different orientations) [7].
May-Britt Moser and Edvard Moser shared the 2014 Nobel Prize in Physiology or Medicine with John O’Keefe for their discoveries of cells (grid and place cells, respectively) that subserve the brain’s internal positioning system. From a modelling perspective grid cells have attracted a lot of attention, due in part to their relatively recent and unexpected discovery, but also due to the very geometric firing patterns that they generate. There are now three main competing mathematical models for the generation of grid-like firing patterns: oscillatory interference models, continuous attractor network models, and “self-organised” models—see Giocomo et al. [8] and Schmidt-Hieber and Häusser [9] for excellent reviews. The first class of model uses interference patterns generated by multiple oscillations to explain grid formation [10]. They have been especially fruitful in addressing the theta rhythmic firing of grid cells (5–12 Hz) and their phase precession. Here spikes occur at successively earlier phases of the theta rhythm during a grid field traversal, suggestive of a spike-timing code [11]. The second class uses activity in local networks with specific connectivity to generate the grid pattern and its spacing. In this category, the models can be further sub-divided into those that utilise spatial pattern formation across the whole tissue (possibly arising via a Turing instability), such as in the work of Fuhs and Touretzky [12] and Burak and Fiete [13], and those that rely only upon spatially localised pattern states (or bumps) in models with (twisted) toroidal connectivity as described by McNaughton et al. [14] and Guanella et al. [15]. The third class proposes that grid cells are formed by a self-organised learning process that borrows elements from both former classes [16–18]. Recent experiments revealing the in vivo intracellular signatures of grid cells, the primarily inhibitory recurrent connectivity of grid cells, and the topographic organisation of grid cells within anatomically overlapping modules of multiple spatial scales along the dorsoventral axis of MEC provide strong constraints and challenges to all three classes of models [18–20]. This has led to a variety of new models, each with a focus on one or more aspects of biophysical reality that might underlie the functionality of grid cell response. For example Couey et al. [21] have shown that a continuous attractor network with pure inhibition can support grid cell firing, with the caveat that there is sufficient excitatory input to the MEC, supposedly from hippocampus, to cause principal cells to fire. However, recent optogenetic and electrophysiological experiments have challenged this simple description [22], highlighting the importance of intrinsic nonlinear ionic currents and their distribution amongst the main cell types in MEC.
Stellate and pyramidal cells constitute the principal neurons in layer II of medial enthorinal cortex (MEC II) that exhibit grid cell firing. The former comprise approximately 70% of the total MEC II neural population and are believed to represent the majority of the grid cell population. Even before the discovery of grid cells stellate cells were thoroughly studied because of their rapid membrane time constants and resonant behaviour. Indeed, they are well known to support oscillations in the theta frequency range [23, 24]. Interestingly the frequency of these intrinsic oscillations decreases along the dorsal-ventral axis of MEC II [25], suggestive of a role in grid field spacing. The resonant properties of stellate cells have been directly linked to a high density of hyperpolarisation-activated cyclic-nucleotide-gated (HCN) channels [26], underlying the so-called \(I_{\text{h}}\) current. The time constant of both the fast and slow component of \(I_{\text{h}}\) is significantly faster for dorsal versus ventral stellate cells, providing a potential mechanism for the observed difference in the resonant frequency along the dorsal-ventral axis. However, perhaps of more importance is the fact the \(I_{\text{h}}\) current can cause a depolarising rebound spike following a hyperpolarising current injection. Given that stellate cells are mainly interconnected by inhibitory interneurons [21], this means that rebound can play an important role in shaping spatio-temporal network rhythms. The inclusion of important intrinsic biophysical properties into a network has been emphasised by several authors, such as Navratilova et al. [27] regarding the contribution of after-spike potentials of stellate cells to theta phase precession, and perhaps most notably by Hasselmo and colleagues for the inclusion of HCN channels [28–31]. This has culminated in a spiking network model of MEC that supports patterns whose periodicity is controlled by a neuronal resonance frequency arising from an \(I_{\text{h}}\) current [1]. The model includes many of the features present in the three classes described above, and is able to replicate behaviour from several experiments, including phase precession in response to a phasic medial septum input, theta cycle skipping, and the loss of the spatial periodicity of grid cell firing fields upon a reduction of input from the medial septum. Simulations of the model in one spatial dimension show that the spacing of grid firing fields can be controlled by manipulating the speed of the rebound response. We note that a change that affects the rebound response would also affect the resonant properties of the cell. In contrast to continuous attractor models that rely on the spatial scale of connectivity to control grid spacing, a change in rebound response provides a mechanism of local control via changes in the expression levels of HCN channels. This fascinating observation warrants a deeper mathematical analysis. In this paper we introduce a spiking network model that shares many of the features of the Hasselmo model [1], focussing on the formation of travelling waves that can arise in the absence of (medial septum) input. Importantly our bespoke model is built from piecewise linear and discontinuous elements that allows for an explicit analysis of the periodic waves that arise in an inhibitory network through rebound spiking. In particular our wave stability analysis demonstrates clearly that the maximum allowed period is strongly controlled by the properties of our model \(I_{\text{h}}\) current. This gives further credence to the hypothesis that HCN channels can control the properties of tissue level periodic waves that may underpin the spacing of grid cell firing in MEC.
In Sect. 2 we introduce our model of a network of stellate cells in MEC II. The single neuron model is a simple leaky integrate-and-fire (LIF) model with the inclusion of a synaptic current mediated by network firing events, and a model of \(I_{\text{h}}\) based on a single gating variable. For ease of mathematical analysis we focus on a continuum description, so that the model may be regarded as a spiking neural field. Simulations of the model in two spatial dimensions are used to highlight the genericity of spike rebound mediated spatio-temporal patterns. To uncover the systematic way in which a cellular \(I_{\text{h}}\) current can control the properties of patterns at the tissue level, we focus next on a one-dimensional version of the model without external input. By further developing a piecewise linear (pwl) caricature of the activation dynamics of a HCN channel we show in Sect. 3 how explicitly to construct the dispersion curve for periodic travelling waves. This gives the speed of a wave as a function of its period, and shows the possibility of a wide range of wave periods that could be selected. Next in Sect. 4 we exploit techniques from the analysis of nonsmooth systems to determine the Evans function for wave stability. Importantly an investigation of this function shows that the maximum allowed period can be controlled both by the overall conductance strength of the \(I_{\text{h}}\) current as well as the time-scale for activation of HCN channels. Finally in Sect. 5 we discuss natural extensions of our work, and its relevance to further models of grid cell firing.
2 The Model
The original work of Hasselmo [1] considered both simple resonant models as well as spiking nonlinear integrate-and-fire Izhikevich units to describe MEC stellate cells. The former incorporated a model for \(I_{\text{h}}\) using a single gating variable with a linear dynamics whilst the latter were tuned to capture the resonant and rebound spiking properties from experimental data. Synaptic interactions between cells in a discrete one-dimensional network were modelled with a simple voltage threshold process. These models were subsequently studied in more detail in [28], with a further focus on two-dimensional models and travelling waves. Here, we consider a biophysically realistic spiking model in a similar spirit to that of Hasselmo, but within a framework that will allow for a subsequent mathematical analysis. In particular, we consider a spiking model of stellate cells using a generalised LIF model that includes a nonlinear \(I_{\text{h}}\) current. Moreover, we opt for a description of synaptic interactions using an event-based scheme for modelling post-synaptic conductances.
An animal’s speed \(v=v(t)\) and direction of motion \(\varPhi=\varPhi(t)\) generates input to the MEC that is modelled by the head-direction current \(I_{\text{hd}}\). For example, this could be of the form \(I_{\text{hd}}(\boldsymbol {r},t)= \boldsymbol {v} \cdot \boldsymbol {r}_{\phi}\) where \(\boldsymbol {v} = v(\cos{\varPhi}, \sin{\varPhi})\) and \(\boldsymbol {r}_{\phi}= l(\cos\phi(\boldsymbol {r}), \sin\phi(\boldsymbol {r}))\) describes a head-direction preference for the orientation \(\phi(\boldsymbol {r})\) at position r [13]. Here l is a constant that sets the magnitude of the head-direction current. The hidden assumption here is that head direction matches the direction of motion. However, this is not necessarily true behaviourally, and head direction cells may not code for motion direction [35]. Nonetheless it is a common assumption in most grid cell models, and so we adopt it here too. Fuhs and Touretzky [12] have shown that choosing an anisotropic anatomical connectivity of the particular form \(W(\boldsymbol {r},\boldsymbol {r}') = w(|\boldsymbol {r}-\boldsymbol {r}_{\phi}-\boldsymbol {r}'|)\) can then induce a spatio-temporal network pattern to flow in accordance with the pattern of head-direction information generated when traversing an environment. For continuous attractor network models that can generate, via a Turing instability, static hexagonal patterns with regions of high activity at the nodes of a triangular lattice, the induced movement of these hot-spots over a given point in the tissue gives rise to grid-like firing patterns. In this case the spacing of the firing field is hard to change, as it is mainly fixed by the spatial scale of the chosen connectivity; however, the mechanism of inducing pattern flow is robust to how the tissue pattern is generated. Thus, given the established success of the Fuhs and Touretzky mechanism we will not focus on this here, and instead turn our attention to the formation of relevant tissue patterns and, in particular, how local control of firing field spacing may be effected.
3 Wave Construction
3.1 Travelling Wave Analysis
4 Wave Stability
To determine the stability of a periodic travelling wave we must not only treat perturbations of the state variables, but also the associated effects on the times of firing. Moreover, one must remember that because the model is nonsmooth (due to the switch at \(V=V_{\pm}\)) and discontinuous (because of reset whenever \(V=V_{\text{th}}\)) standard approaches for analysing smooth dynamical systems cannot be immediately applied. Nonetheless, as we show below, the wave stability can in fact be explicitly determined. We do this by constructing the so-called Evans function. This has a long history of use in the analysis of wave solutions to partial differential equations, dating back to the work of Evans on the stability of action potentials in the Hodgkin–Huxley model of a nerve fibre [39], has been extended to certain classes of firing rate neural field model [40], and is developed here for spiking neural fields.
5 Discussion
This paper is motivated by recent work in computational neuroscience that has highlighted rebound firing as a mechanism for wave generation in spiking neural networks that can underlie the formation of grid cell firing fields in MEC [1]. We have presented a simple spiking model with inhibitory synaptic connections and an \(I_{\text{h}}\) current that can generate smoothly propagating activity waves via post-inhibitory rebound. These are qualitatively of the type observed in previous computational studies [36, 37], yet are amenable to an explicit mathematical analysis. This is possible because we have chosen to work with piecewise linear discontinuous models, and exploited methodologies from the theory of nonsmooth systems. In particular we have exploited the linearity of our model between events (for firing, switching, and release from a refractory state) to construct periodic solutions in a travelling wave frame. To assess the stability of these orbits we have treated the propagation of perturbations through event manifolds using saltation operators. Using this we have constructed dispersion curves showing a wide range of stable periods, with a maximum period controlled by the time-scale of the rebound current. This gives further credence to the idea that the change in grid field scale along the dorsal-ventral axis of the MEC is under local control by HCN channels.
A number of natural extensions of the work in this paper suggest themselves. We briefly outline them here, and in no particular order. For simplicity we have focussed on the analysis of waves in a homogeneous model with only one spatial dimension. The analysis of the corresponding travelling waves, with hexagonal structure, in two spatial dimensions is more challenging, though is an important requirement for a complete model of grid cell firing. Moreover, the assumption of homogeneity should be relaxed. In this regard it would be of interest to understand the effects of a heterogeneity in the time constants (for voltage response, synaptic time-scale, and the time-scale of the \(I_{\text{h}}\) current) on the properties of spatio-temporal patterns. Furthermore, it would be biologically more realistic to consider two sets of inhibitory interneurons, as in [1, 28, 30]. As well as depending on the \(I_{\text{h}}\) current, grid field spacing changes as a function of behavioural context. This is believed to occur through the activation of neuromodulators [32], and simple regulation of our \(I_{\text{h}}\) current model would allow a systematic study of this, even before considering the more important issue of structured input. The work in this paper has focussed on spontaneous pattern formation that occurs in the absence of such input. Given the importance of the head-direction input for driving grid cell firing fields it would be natural to consider a mathematical analysis for the case with \(I_{\text{hd}} \neq0\). For the standard Fuhs–Touretzky mechanism of inducing firing patterns this would further require the treatment of an anisotropic interaction kernel with a dependence on a head-direction preference map. One way to address this would be via a perturbation theory around the limiting case treated in this paper, and use this to calculate a tissue response parametrised by the animal’s speed and direction of motion. The same methodology would also allow an investigation of phase precession during a grid field traversal. Our simulations have also shown the possibility of ‘lurching waves’ and it would be interesting, at least from a mathematical perspective, to analyse their properties (speed and stability) and compare them to the co-existing smoothly propagating waves. It would further be pertinent in this case to pay closer attention to any possible nonsmooth bifurcations that could give rise to wave instabilities, such as grazing bifurcations. Finally we note that grid cells are grouped in discrete modules with common grid spacing and orientation [20]. It has recently been suggested that coupling between modules or via feedback loops to the hippocampus may help to suppress noise and underpin a robust code (with a large capacity) for the representation of position [45]. Another extension of the work in this paper would thus be to consider the dynamics of networks of interacting modules, paying closer attention to the details of MEC microcircuitry [46].
Declarations
Acknowledgements
SC was supported by the European Commission through the FP7 Marie Curie Initial Training Network 289146, NETT: Neural Engineering Transformative Technologies. We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Tesla K40 GPU used for this research. KCAW was generously supported by the Wellcome Trust Institutional Strategic Support Award (WT105618MA). MBQ gratefully acknowledge the support of CONACyT and The University of Nottingham during her PhD studies. We thank the anonymous referees for their helpful comments on our manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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