The motivation for a novel theory
5.4 Discussion and remarks on the biological plausibility
Proof. The corresponding graph coloring problem introduced in Section 5.2 is used to proof the theorem in two dimensions. The densest arrangement of spatial symbols is depicted in Figure 5.2b and forms a hexagonal lattice. Transitions between symbols are only possible between adjacent symbols. In other words, all symbols δj that are at most 2rw apart from δi form a local transition group. Consequently, the corresponding transition graph extracted according to the method described in the proof for Corollary 2 is not complete, i.e. only the local transition group is connected.
The transition graph for two dimensions is depicted in Figure 5.3a. The chromatic number of the resulting graph is 3 and the occurrence of colors is periodic as depicted in Figure 5.3b.
It is conjectured that the proof will hold in higher dimensions.
5.4 Discussion and remarks on the biological plausibility
MTT defines symbols and transitions both for purely episodic as well as spatial information, the first in case of a universal MTS. Likely neural candidates for symbols are place cells of hippocampal area CA3. As these cells are relevant for both episodic as well as spatial information processing [229, 251, 265, 319], it is proposed that they form encoders of spatio-temporal symbols. Similar to the model by Barry et al. [13], it is proposed that place cells form primarily on spatial afferents. However, it is further proposed that the meaning of a symbol in a universal MTS, and hence of a place cell, can be extended such that it integrates additional non-spatial afferents due to the observation of conjunctive place cells [225, 247]. The preceding analysis of a MTS remains unchanged by this extension.
So far it is unclear if inter-neurons in CA3 could represent temporal transitions, or if temporal transitions are rather encoded in the place cells of CA1. Recurrent collaterals in CA3 are in favor of the first assumption [11, 205]. Recurrent collaterals from CA1 to CA3 have not been described often enough in support of the second possibility. However, it is possible that the recurrent connectivity is not mono-synaptic but is represented by what is known as the trisynaptic loop, spanning from CA3 across CA1 to EC, before it arrives back at CA3 [6].
Following MTT and Theorem 2 in particular, grid cells are proposed to represent spatial information in form of transitions between spatial symbols. It is proposed that they not only form on the basis of pre-synaptic spatially modulated input, but also affected by recurrent connectivity from place cells. Both propositions were already observed in real grid cells [28, 55]. From a computer scientific perspective, this setup of interactions can be considered an abstraction layer and will be explored in detail in Chapter 7. To anticipate, place cells are suggested to form a storage mechanism for arbitrary points of a sequence, whereas grid cells encode their spatial transitions and provide spatial neighborhood information of places. In this way, place cells are unaware of spatial relations except via the indirection of grid cells. Consequently, the sensory representation by which grid cells formed in the first place may change over time, but the spatial relationship, and therefore knowledge of potentially neighboring locations, is maintained.
The spatial MTS Lrequires a unique sensory representation to identify singular locations and for the optimal sampling assumption. Furthermore, it is necessary to
detect the change between states. Candidates for a neural representation of such a signature and change between locations are head direction, tactile information, optic flow or generally speaking distal visual cues, and ego-motion. Optic flow was successfully used not only to model grid cell firing characteristics in an oscillatory inference model [291, 292]. It was shown to be sufficient to account for boundary cell responses as well [294]. The latter finding could explain observations in real recordings of grid cells in which they were influenced by the geometry of the environment [198, 199]. Furthermore, optic flow contains sufficient information to extract ego-motion which, in turn, is represented in MEC by speed cells which fire linearily with respect to the animal’s speed [195]. Additionally, boundary vector cells were successfully used to encode locations and drive place cell activity in a computational model and thereby discriminate positions [13]. Finally, sensory cues were able to stabilize continuous attractor dynamics in a network model of grid cells [258]. Interestingly, it was reported that grid cells require visual input for their periodic responses [55]. It is thus conjectured that visual input and the boundary vector state provide sufficient information for the encoding of locations and formation of transitions in experimental environments of two and three dimensions.
The optimal sampling process requires a dense representation of the input space.
Hence, the tuning curves of neurons pre-synaptic to grid cells are expected to overlap appropriately. It is known from several cortical areas, especially the auditory and visual cortices, that neurons show overlapping tuning curves which are indeed well separated, uniformly cover the input space, and are often organized topographically [34, 95, 157, 339]. In all studies, the amount of overlap depends on the tuning width of the neurons and the number of neurons employed to sample from the space.
On the other hand, the amount of overlap of grid cell firing fields is expected to decrease over time. Consider two adjacent spatial symbols at locations xi, xj in a continuous one dimensional spaceD, encoded in form of two neurons such that their receptive fields cover the distance, i.e. rs= 2rw. Then, the precise relative spike time of the two neurons contains sufficient information to determine the exact location between xi and xj. When does the transition appear and how to encode it? One may suggest that the transition occurs when the difference between spiking times of the neurons changes sign, i.e. when the location moves from rw of one neuron to the next. However, the transition will get activated throughoutrs, and thereby violating the coherency constraint. Therefore it is suggested that a process exists which will try to maximally separate the tuning curves of the cells depending on experience within an area. Then, the activation of a spatial transition neuron would initially start to correspond with rs but shrink to rw over time. An effect which was already observed in real recordings from the EC [14, 15]. As a by-product, the response field of a transition neuron will likely reduce to a Voronoi cell. Hence, perfect rotational symmetricity with respect torw will decrease. Another feasible solution to the constraint violation is to associate a transition bundle only with targets outside ofrs, and ignore any other symbols co-active within rs. Other transition cells would then be required to densely cover the set of spatial symbols. Local self-organization principles, for instance such as suggested in the computational model for grid cell formation by Kerdels et al. [178], or attractor dynamics could lead to a coherent representation within one grid module.
Besides preventing violation of the coherency constraint by transition points, the
5.4 Discussion and remarks on the biological plausibility 49
network has the ensure the unambiguity of the transition bundle given its sensory information. This means that at any location which is uniquely identifiable by sensory information, only the transition bundle which is associated with the corresponding sensory state is allowed to be active. This suggests that the transition network is governed predominantly by local inhibitory recurrences generating a winner-take-all mechanism. In fact, it was already observed that recurrent connectivity in mEC is primarily inhibitory [70]. The local inhibition is required to be fast enough to prevent erroneous activation of a transition unit. Such a temporally quick effect has been observed already in the Hippocampus [86], and it is expected that local recurrences in mEC are equally fast.
A model which yields the general behavior expected from the presented theory was published recently by Widloski et al. The authors proposed a spiking neural network which was driven by spatially modulated input and formed a hexagonal lattice in two dimensions [379]. However, the computational necessity of grid cells was not addressed in the work.
Conclusively, the prerequisites for neural implementations of both temporal and spatial MTS exist. A biologically plausible model for grid cells in continuous metric space is derived in Chapter 6. The algorithmic interactions between temporal and spatial MTS are the subject of Chapter 7.