What We Know About Grid Cells

Since their Nobel Prize winning discovery in 2005, grid cells have been studied extensively. Their multi-scale periodic firing rates tiling the environment as the animal moves around has been shown as critical for path integration. Multiple experiments have shown that grid cells also fire for other representations such as olfactory, attention mechanisms, imagined movement, and concept organization showing the possible mechanism for cognitive maps. In this paper we give an overview of grid cell research since their discovery, their role in neuroscience, cognitive science, and artificial intelligence research, and possible future directions of artificial intelligence research.

The Entorhinal cortex, also called the medial entorhinal cortex (MEC), is an area in the brain that sits between the neocortex and hippocampus in an area called the medial temporal lobe. In Broadmann’s area, it is Broadmann area 28. Ramon y Cajal described the area in 1902, but at the time thought it was part of the olfactory system used for processing smell information.

In 2005, grid cells were discovered in the entorhinal cortex [hafting-2005]. Grid cells are a type of neuron that fires at regular intervals as an animal navigates an area. Grid cells provide a multi-scale periodic representation that functions as a metric for location encoding and is critical for recognizing places for navigation. When the scientists plotted the points in 2D, they discovered that they formed a grid of tesselating triangles, hence the name grid cells.

The team that discovered grid cells consists of Edvard Moser, May-Britt Moser, and their students. The team ending up winning a Nobel peace prize in Physiology or Medicine in 2014. Coincidently, they shared the prize with John O’Keefe who discoverd place cells in the hippocampus. Together this Hippocampus and Entorhinal cortex (HEC) system are believed to be the main system for building a cognitive map. The HEC system is also one of the most frequently studied areas connecting neuroscience with artificial intelligence.

Neuroscienists have been optimistically studying the HEC system trying to unlock how exactly it works to accelerate various fields of science. Place cells in the hippocampus are special in the brain in that their firing rates act as bookmark for landmarks or an indexing system. Tthey fire only foor specific locations in an environment. For example a particular place cell may only fire when at the entrance to work, your favorite parking spot, or at your house. Place cells are known to fire on 1 to several locations. Contrast that with grid cells, they retain their basic firing pattern regardless of environment, making them generalized and context independent. The firing rate forms a hexagonal like grid pattern that tiles to the environment. Grid cells and Place cells feed into each other along with other inputs forming some kind of recurrent network.

Grid cells have been found in mice, rats, bats, monkeys, and humans. It is believed that they exist in all mammals similar to the neocortex. It is still unknown if grid cells appear in non-mammal animals.

There are 3 parameters that determine a grid cell’s firing pattern, the orientation of the grid, the spacing between firing fields (also called the wavelength), and the location (also called spatial phase) of the grid fields. Grid cells cluster together into idscrete modules sharing orientation and periodicity, but varying randomly in phase. Currently for all possible configurations of parameters, it is believed that there are under 10 different modules. Grid cells will maintain their firing patterns even if speed or direction change. Grid cells are usually clustered together forming discrete modules of computation.

In terms of morphology, both stellate cells and pyramical cells can be grid cells.

To navigate and interact with our world, our brains must store this information in a way that we can retrieve it and use it in new situations. One interpretation of the brain is that it is a coordinate translation machine: "a brain is a well-designed machine for the frame conversion to internalize the external world" [arisaka-2022] and "the egocentric representations of the primary sensory cortical areas must be transformed into an allocentric representation in the hippocampus, and then transformed back to an egocentric motor representation for behavioral output" [byrne-2009]. The brain has been found to store multiple coordinate representations along with different orientations or points of view, mainly allocentric and egocentric orientations.

Grid cells have an allocentric point of view (i.e. world-centered, external world, frame of reference). Their firing does not encode data from the point of view of the agent or ego. On the contrary, egocentric neurons fire in relation to the animals point of view such as visual neurons. A 3D first person shooter is played from the ego’s point of view where as tic-tac-toe is played in an allocentric manner, you can the play game from just knowing the environment coordinates and in any orientation. In the HEC, it is thought that most of the representations are allocentric, but there are new studies showing that egocentric representations may also be stored. In one study,Kutz et al. found abundant cells in the parahippocampal cortex of human epilepsy patients that they dubbed "anchor cells". These anchor cells seem to represent egocentric direction towards "anchor points" located in the environment[kunz-2020]. It is thought that these anchor cells are used to transform perceptual signals into allocentric representations.

Navigation is crucial for animals to survive in nature. All animals must learn to navigate their environment to forage, mate, exploring, and return home. They must be aware of their allocentric position by integrating their self-motion, other somatosensory information, and enviromental cues. This process is known as path-integration and grid cells in mammals are crucial to this process.

The entorhinal cortex is made up primarily of grid cells, but there are other functional cell types as well: head direction cells (HD cells), boundary vector cells (BVCs), and speed cells. There are also cells that form combinations of these cells. Other cell types continue to be discovered.

Grid cells mainly feed into place cells, but its not that simple. They both feed into eachother forming a recurrent network. Experiments have shown that place cells develop before grid cells [langston-2010]. Several studies have shown MEC neurons receive input from hippocampal place cells: Place cells in the hippocampus emerge earlier during development than grid cells in the mEC (Langston et al., 2010; Wills et al., 2010), grid cells lose their tuning pattern when the hippocampus is deactivated (Bonnevie et al., 2013) and both the firing fields of place cells and the spacing and field size of grid cells increase along the dorso-ventral axis (Jung et al., 1994; Brun et al., 2008b; Stensola et al., 2012). Moreover, entorhinal stellate cells, which often exhibit grid-like firing patterns, receive a large fraction of their input from the hippocampal CA2 region (Rowland et al., 2013), where many cells are tuned to the location of the animal (Martig and Mizumori, 2011). In one paper, researchers propose that place cells feed into grid cells to act as a lower dimensional reprsentation of place cells in a principle component analysis (PCA) form [dodek-2016]. Some experimental evidence has shown that place cell input to grid cells acts as a correcting mechanism for drift and anchors grids to environmental cues [barry-2007]. Another interpretation is that grid cells may be understood as the eigenvectors of the relationship between place cells. Another interpretation of the interplay between grid cells and place cells is that grid cells act as a coding schema or lookup table for accessing specific memories stored inside of place cells. In another paper, Bush et al. ran experiments that show that grid cells to place cells firing is not a successive processing pipeline, but instead that place cells are taking multiple inputs from different areas and that grid cells provide a complimentary and interacting signal to support reliable coding of large-scale space [bush-2014]. Our understanding of the interaction of grid cells and place cells is still primitative and much more experiments need to happen.

Spatial navigation is thought to be the main focus of grid cells and the entorhinal cortex, but with its generalized grid code that fires consistently regardless of the environment, many have wondered if this grid code is used to represent other neural representations. In the following paragraphs we will describe experiments that focus on grid codes that represent other sensory signals and representations. The results of these experiments seem to show that grid cells may have evolved for other uses acting as a form of neural recycling.

To go from one location to another requires movement which must happen over time. Time and space are deeply intertwined, for example in physics, spacetime is a mathematical model that merges the three dimenions of space with one dimension of time into a single 4D manifold. Scientists have wondered if grid cells also represent time. It has been known that cells in the hippocampus can represent abstract time directions [eichenbaum-2014]. In a study from Kraus et al., the group of researchers studied rats in a treadmill running at various speeds and distances. They built several models of the data to factor out location and found that the grid cells were actually weakly association with location. They found that most grid cells where either strongly correlated with a combination of time and distance or either just time or distance. Their findings suggest that in the absence of external queues, grid cells integrate self-generated distance and time information and that grid cells with the hippocampus represent both temporal and spatial dimensions.

The majority of grid cells studies has been on 2D physical spaces. All animals are moving in a 3D space, especially animals that are fly or live in the sea such as bats and dolphins. So the next natural step has been to understand grid cells in 3D. Researchers expected the same periodic firing pattern in a lattice like structure to appear while traversing 3D space. Place cells and head direction cells have already been shown to directly generalize to 3D. Many experiments have been run to understand 3D grid codes, but while 2D representations of grid code is undeniable, finding 3D representations has remained an elusive problem. In a paper from Ginosar et al. [ginosar-2021] they recorded MEC grid cells of flying bats to understand 3D grid code patterns. The team created an enclosure of size 5.8m x 4.6m x 2.7m with several feeding areas placed throughout the enclosure to encourage the bats to fly to different locations. They expected the hexagonal grid code to move uniformally into 3D like a stack of oranges in a grocery store representing optimal packing, but the results show that in 3D there was no neat uniform pattern. They claim their results show that in 3D, many cells have multiple 3D fields and that in 3D the grid code has local structure and no global lattice, but when constrained to 2D will form a hexagonal global lattice. Given the findings, they suggest that global spatial periodicity is not central to grid cells. In another paper [grieves-2021], researchers tested 3D grid cells with rats exploring a 3D space. Their results are similar and found that the grid fields were irregulary arranged, sparser, large, and more variably sized. They suggest that grid codes may not need to be tiling or tightly packed to support spatial computations. Therefore new models need to be created to describe grid cell hippocampus interactions. There are several computational and mathematical models that have been developed that do fit to grid cell firing patterns in both 2D and 3D. If we are to get to a multidimensional abstract conceptual space using grid cells, then more 3D and multidimensional experiments with time will need to be performed.

The majority of grid cell experiments done on humans is used with functional Magnetic Resonance Imaging (fMRI). fMRI is used most often to measure blood-oxygen-level-dependent (BOLD) signals, or in other words changes of oxygen levels in the blood as a proxy for neuron activity. It shows where blood is being sent in the brain, presumably because neurons in that area are more active during a mental task. It cannot directly measure neuron communication through electric or chemical impulses. It is noninvasive which means its can be used on practically any human. From a technical point of view, it has multiple problems. The fMRI acquisition time is around 2 seconds to perform one measurement and this blood flow is only correlated with increased neuronal activity. This is eons compared to the speed of computation that neurons communicate at. To give an analogy, its like measuring how a boat engine functions by measuring distant waves the boat produces as it moves. fMRI resolution is measured in voxels which is a 3d pixel that ranges anywhere from 1 to 5 cubic millimeters in size. A voxel contains approximately 1 million neurons. When fMRI measurements are occuring, the patient must remain completely still, otherwise it will throw off the measurements. After the raw data is acquired, mathematical analysis must occur to interpret the data which can lead to other types of statistical problems. In a real, but satirical study titled "Neural Correlates of Interspecies Perspective Taking in the Post-Mortem Atlantic Salmon: An Argument For Proper Multiple Comparisons Correction", a dead salmon was placed in an fMRI machine and then showed pictures of humans in different emotional states. The authors were able to show the dead salmon had significant increases in "activation" highlighting the importance of correcting your statistics in fMRI studies [bennet-2012]. Only when there are patients with severe problems such as epileptic patients can direct measurement of neuronal activity occur in humans. So a large poriton of grid cell experiments continues to happen on rodents and other mammals. Of note, these measurement problems apply to the majority of human neuroscience experiments, not just grid cell experiments. If we are to improve our undertanding of grid cells and the brain, much more precise and improved tools will need to be invented to examine neuronal communication.

To calculate if a particular neuron fire for grids, often times a gridness score is used [sargolini-2006]. The gridness score is calculated by rotating the firing map in increments of 30 degrees. For every rotated angle, the pearson correlation is compared with the original un-rotated data.

The EC seems to be a critical part of the mind and brain and its degradation seems to have severe effects on human cognition and health. In Alzheimer’s disease, the EC is one of the first areas of the brain to be noticeably effected, typically with a reduction in volume of the cortex [vanhoesen-1991]. Early changes in the EC are used as a strong predictor of Alzheimer’s disease [bottero-2021] . Patient HM (1926-2008) was the most studied patient in neuroscience history [squire-2009]. He had seizures since 10 years old from an accident and at 27 years old had a surgery that removed his hippocampus, entorhinal cortex, and nearby areas. The procedure stopped his seizures, but it created another problem in that he acquiried anterograde amnesia, he was unable to form new memories. At the time of his death grid cells were only recently discovered and so there were not many experiments done with him, but the fact that his EC was missing implicates it as being a key system for declaring new memories.

The lure of a new type of computational processing unit that may work for both spatial and non-spatial snsory processing has computer scientists excited. AI researchers have been increasingly investigating grid cells to incorporate into artificial neural networks (ANNs). There have been several studies trying to incorporate grid cells in various ways and we will review some of the most interesting papers.

In a paper from [banino-2018], researchers tested out path integration using a RNN that had cells that fired and looked just liked grid cells. The authors built a deep learning LSTM model trained on locations, velocity, and head direction to predict where the agent would move to. They used spatial autocorrelograms to analyze the firing rates of the artificial neurons and found that the LSTM produced cells that behaved just like hexagonal grid cells and BVCs. They then took the LSTM model and used it as a base layer for a A3C reinforcement learning agent that navigated an environment. They found that the RL agent was able to navigate and take previous unseens paths faster than all the models they compared it to. Of note, they were only able to produce grid like firing patterns when they incorporated dropout. In our own experimentation, we were only able to get grid patterns when we trained at the same batch size of 10 that they did, anything else higher drastically reduced the amount of grid cells appearing. The findings suggest that grid cells are a natural computational unit that will appear naturally in navigational problems. Deepmind has subsequently went on to patent this computational pattern in "Performing navigation tasks using grid codes" [banino-2021].

In another paper that builds off the work of [banino-2018], researchers wanted to see what was the minimal neural network architecture needed to make grid like cells appear. They tested several different architectures ranging from a 1 layer NN, LSTM, to RNN and found that all of those models were able to produce grid like patterns IF they followed certain constraints. They found the minimal needed architecture is "a simple linear, place cell encoding objective" with non-negative constraints. They also found that the shape of the grid pattern depends specifically on the shape of the cell tunng curve. They found their findings confusing seeing that several previous paper needed much more to make grid cells appear [sorcher-2019].

In another paper, a team of researchers implemented grid cells in an ANN to study how a network may process arbitrary sequence of input samples similar to how bilogical eyes use saccades [2021-leadholm]. They make the case that ANNs can only handle sequential processing while humans handle out of sequence recognition easily and naturally. They explored grid cell path finding mechanisms for their model and found that it performed similarly to a CNN when tested on the MNIST dataset.

The Transformer architect is currently one of the fastest growing innovations in ANNs and have made many breakthroughs in language modeling, image processing, and controlling computers. In a paper from Whittington et al., they have shown that when a small modification is made to the transformer architect, they learn and act like grid cells and place cells [whittington-2022]. They modify the transformer architect by adding recurrent positional encodings which creates a form of path integration. The core mechanisms of transformers is selt-attention where each element is trained to attend to all the other elements and update itself accordingly. Their findings help make a stronger link between neuroscience and machine learning and a potential roadmap for further exploration in machine learning grid cell integration.

There have been multiple experiments that show grid codes similar to grid cells exist in other parts of the brain such as the neocortex. In the paper "Evidence for grid cells in a human memory network", They wanted to see if they could find grid-like firing using fMRI signal, which reflects changes in metabolic activity across large portions of the brain resulting in very coarse readings of neuronal activity. They designed an experiment using a virtual environment in a way that they could measure 3 factors from the fMRI data. 1) The angular orientation relative to the environment, 2) The running direction, and 3) The running speed. They believe they did find grid-like cell activity in multiple neocortex regions such as the visual cortex and retrosplenial cortex. They note “Because we ca only measure effects of direction and speed (not location) in fMRI signal, our findings could reflect the presence of grid cells, or movement-related responses from head direction, or ‘conjuctive’ directional grid cell…" In another paper, "Direct recordings of grid-like neuronal activity in human spatial navigation", researchers surgically implanted electrodes into epilepsy patients. They had recordings for 893 individual cells in the hippocampus, amygdala, parahippocampal gyrus, EC, and cingulate cortex. They had those patients perform navigation tasks in a virtual environment cut up into a 28 x 28 array. They came up with a gridness score to classify if the neurons had grid-like codes. Their results found that of all the recorded cells, 14% of cells in EC, 12% of cells in the cingulate cortex, and 8% of the hippocampus cells had a strong gridness score.

In a paper by Constantinescu et al, They designed an experiment for humans to navigate through an abstract conceptual presentation consisting of “bird space”, arrangements of different lengths of bird necks and legs. They then measure the participants over several sessions spanning weeks using fMRI. In the fMRI signal, they found hexagonally symmetric activity in the vmPFC.

In another paper by Long et al, scientists measures 2025 cells from 8 implanted rats that were able to freely move. They did 287 recording sessions of their neuron firing activity. They found that cells known primarily to be in the entorhinal cortex such as head direction cells and border vector cells, were found in the somatosensory cortex. Of all their cell recordings, they found 3.55% fired like grid cells. They found that these grid cells were a little nosier and less prevalent than the other cell types they found. They hypothesize that this may be due to other factors such as proprioception.

With multiple experiments showing grid codes in other parts of the brain, it seems like grid like navigation code has multiple uses for intelligence. Hawkins et al. believe that the neocortex uses grid cells as a fundamental part of human level intelligence and are trying to build new computer algorithms that use neocortical grid cells [hawkins-2019].

Scientists have discovered MEC neurons such as grid cells in other regions of the brain. It is thought that grid cells may appear in more regions of the brain, but it is difficult to detect when not explicitly looking for grid-like responses. Specific grid cell experiments must be constructured when looking for grid responses. Moreover, Many neurons in the neocortex and somatosensory regions seem to respond to more diverse stimulus making it hard to analyze.

We hope that this overview of grid cells helps you understand their importance to neuroscience, cognitive science, and artificial intelligence. For neuroscience it gives researchers a window deep into the cognitive processing pipeline that allows us to experiment and measure directly inputs to outputs. In a paper from Jonas et al. titled "Could a Neuroscientist Understand a Microprocessor?"[jonas-2017], they argue that the complexity of the brain and the crude tools to run experiments often produce results that are not meaningful towards the understanding of neural processing systems. They show this by using neuroscience techniques to analyze a human engineered microprocessor and the results are that they could not meaningfully understand its information processing systems. With grid cells, we have a method to measure directly part of the information processing pipeline giving us a direct mapping of movement inputs to grid code outputs. For cognitive science, its generic code along with all the other MEC cells show that the brain does have basis cells for computation that lead to other computations in the hippocampus and other places.

In artificial intelligence research, artificial neural networks (ANNs) have made huge progress in the past decade with summation point neurons as the fundamental building block. Some question whether the trend of "just scaling up" by adding more computation and parameters will lead to more intelligent models. A neuron on average transmits action potential or operations at the rate of 1000/sec while a CPU can do 10 billion/sec. The method and efficiency of the way human brains process information versus how computers process information is fundamentally different. The past few decades of AI research has shown that pure computer science based AI research has led to many dead end paths such as expert systems, ontologies, symbolic AI, among others. If we are trying to emulate human level intelligence, then studying human brains seems very reasonable. Demis Hassabis has been advocating this approach of neuroscience-inspired AI to reach the next level of intelligent machines and they have taken this approach at DeepMind [hassabis-2017]. Many innovations in deep learning were inspired from neuroscience findings such as convolutionl neural networks (CNNs), point summation neurons, and attention mechanisms. Grid cells offer the potential to be another building block to build new types of ANNs in a wide variety of applications.

Its abstract hexagonal firing code continues to be studied to understand its relation to other cognitive processes such as language comprehension and concept formation as a form of neural recycling. With grid cell experiments showing they can fire for organizing concepts into a conceptual space, there are many possible experiments that could be explored integrating grid cells into ANNs. There are already multiple experiments attempting to converge grid cells and machine learning in ineresting ways. For researchers trying to advance the state of the art in ANNs, grid cell firing codes are a prime target to investigate.

Written by Jason Toy (@jtoy)