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Edited by: Yu-Guo Yu, Fudan University, China

Reviewed by: Shenquan Liu, South China University of Technology, China; Bailu Si, University of Chinese Academy of Sciences (UCAS), China; Lianchun Yu, Lanzhou University, China

This article was submitted to Neuroenergetics, Nutrition and Brain Health, a section of the journal Frontiers in Neuroscience

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Place cells are important elements in the spatial representation system of the brain. A considerable amount of experimental data and classical models are achieved in this area. However, an important question has not been addressed, which is how the three dimensional space is represented by the place cells. This question is preliminarily surveyed by energy coding method in this research. Energy coding method argues that neural information can be expressed by neural energy and it is convenient to model and compute for neural systems due to the global and linearly addable properties of neural energy. Nevertheless, the models of functional neural networks based on energy coding method have not been established. In this work, we construct a place cell network model to represent three dimensional space on an energy level. Then we define the place field and place field center and test the locating performance in three dimensional space. The results imply that the model successfully simulates the basic properties of place cells. The individual place cell obtains unique spatial selectivity. The place fields in three dimensional space vary in size and energy consumption. Furthermore, the locating error is limited to a certain level and the simulated place field agrees to the experimental results. In conclusion, this is an effective model to represent three dimensional space by energy method. The research verifies the energy efficiency principle of the brain during the neural coding for three dimensional spatial information. It is the first step to complete the three dimensional spatial representing system of the brain, and helps us further understand how the energy efficiency principle directs the locating, navigating, and path planning function of the brain.

The spatial cognition function is one of the most important functions of the brain. Many types of cells are contributing to the locating and navigating function. Among them, place cells in hippocampus and grid cells in entorhinal cortex are the most fundamental and well-studied cells. The colorful researches during the last few decades have revealed the representational function of place cells in the hippocampus. Spatial receptive fields of spiking neurons in the rat hippocampus are firstly reported by O'Keefe and Dostrovsky (

Since the discovery of place cells, many models have attempted to explain how this spatial selectivity arises within the hippocampus (Samsonovich and McNaughton,

The spatial representation of place cells is essentially a neural information coding problem, which has been the core problem in cognitive neural science (Amari and Hiroyuki,

Due to the defects mentioned above, it is quite necessary to study the three dimensional spatial representation function of place cells by the energy method. In this research, we constructed a network model for place cells to represent three dimensional space on an energy level. The cells which have various place fields achieved accurate locating function. Then we analyzed the energy consumption properties and locating errors under the situations of different field sizes. The results have shown that this model captured the basic behaviors of place cells and revealed the energy efficiency property of the neural system.

It is very difficult to directly measure the energy consumption of a cell due to the limitation of current recording techniques. However, it is possible to calculate the energy consumption of a cell based on a proper model describing the ion currents (Laughlin et al.,

The equations of H-H model are:

where _{m} is membrane capacitance of a neuron, _{m} is membrane potential, _{Na} and _{K} are Nernst potentials of Na^{+} and K^{+}, and _{l} is the potential while there is no leakage current. _{l}_{Na}, and _{K} are, respectively, the leakage conductance, Na^{+} channel conductance, and K^{+} channel conductance. The typical values of these parameters are: resting membrane potential _{r} = 67.3 mV, maximum Na^{+} conductance _{Na} = 120 mS/cm^{2}, maximum K^{+} conductance _{K} = 36 mS/cm^{2}, leakage conductance _{l} = 0.3 mS/cm^{2}, and Nernst potentials are 50, −80, and −56 mV, respectively. Based on H-H model, we can theoretically calculate the energy consumption of neuronal activity. The energy consumed by a neuron during a certain period of time can be deduced. The equation is shown as follows (Laughlin et al.,

Then it can be calculated that ~1.88 × 10^{−7} J energy is consumed by a typical neuron during an action potential (Wang et al.,

We set up a cube space with a side length of L (see Figure

Simplified three dimensional space model and the neural network.

In the experiment, the environment is a cube place without any references other than the borders, where the animal can only get visual or auditory information from six walls (landmarks). We refer to the six walls as front (F), back (B), left (L), right (R), up (U), and down (D). When study the locating function of place cells in two dimensional space, researchers (O'Keefe and Burgess,

Note that the actual location of the animal at time t is described by vector X(t) = (x_{1}(t), x_{2}(t), x_{3}(t)), where x_{i}(t) is the actual distance from wall

where α represents the error rate of the sensory (visual or auditory) perception, which is dependent on the individual animal. η is a random number from a uniform distribution within the interval [−1, 1], that is η~U(−1,1) (Yan et al.,

The place cells are considered to receive the geometric inputs of boundary vector cells, each of which responds when a boundary is at a particular distance to the animal (Hartley et al., _{ij}(t)]_{3×N}, where _{ij}(t) is the connection weight from the ith sensory neuron to the jth place cell at time t. Weights are initialized by the following functions (Kulvicius et al.,

Where γ is a random number uniformly distributed on the interval [0, 1], that is γ ~ U (0, 1).

Initial weights distribution.

This is a histogram of the initial weights distribution. The horizontal axis represents the synaptic strength, and the vertical axis represents the number of synapse with the corresponding strength. Since γ is uniformly distributed and Equation (4) is central symmetric, the distribution of synaptic strength is symmetric about 0.5. Such a distribution rather than a uniform distribution is that all place field centers will be located around the center of the environment and the model will fail to obtain place fields near the boundary of the environment if the uniform distribution is applied (Kulvicius et al.,

Inspired by the firing rate model in two dimensional space proposed by O'Keefe and Burgess (

Where _{j}_{m} is the maximum firing rate of a single place cell, which is about 20 Hz (Hartley et al., _{j}_{j} is a random number from a normal distribution which initially affects the range of place field. As a result, the random number σ_{j}~N (0.03, 0.005) is another parameter to reflect the diversity of place cells.

According to a classical learning rule with a winner-takes-all mechanism, the connections to the cell with the maximal firing rate wins the learning chance (Kulvicius et al.,

Where μ is the learning rate, _{thr} is the responding threshold represents the minimum firing power of cells that are activated.

The firing power of place cell _{thr}. After firing powers are calculated, place field centers can be defined by analyzing the positions within the corresponding place fields. Furthermore, the center of place field related to cell

Then the location of the animal will be estimated by the weighted average of place field centers according to the response set:

Where, _{j} is the place field center of cell _{j}

According to the described method, we calculate the neural energy consumed by place cell firing an action potential and perform the numerical simulation first.

Figure ^{+}, K^{+}, leakage, and stimulus currents, respectively. By integrating the power over time, we can get that one action potential costs about 188 nJ energy.

Energy consumption by a place cell during an action potential (Wang et al.,

The exploration is performed in three dimensional space by the model. The size of the cube space is 20 × 20 × 20 units, number of place cells is 200, number of sensory neurons is 3, sensory error rate α = 0.1, learning rate μ = 0.001, maximum firing rate R_{m} = 20 Hz, and P_{thr} = 0.3 P_{m}, where P_{m} is the maximum firing power. During the experiment, the animal initiated the random search in the cube space. The number of steps is set to be 10,000, and step length is 1. The trajectory of one random search is shown in Figure

Random exploration trajectory in three dimensional space.

After the spatial exploration and learning, place cells activities are tuned to specific spatial locations to form the place fields.

Figure

Firing powers of place cells in three dimensional space.

Distributions of maximum firing power and size of place field.

Normally the place field centers are near the locations with maximum powers. Two hundred place field centers are summarized in Figure

The distribution of place field centers in three dimensional space.

Figure

Spatial selectivity of place cells.

The various spatial selectivity and the corresponding place fields indicate that the model simulated the basic features of place cells in three dimensional space. As soon as the place fields are formed and field centers are defined, the locating function of the network can be performed. As described in section Energy Model for Place Cells and Learning Rule, the location of the animal will be determined by the weighted average of place field centers belong to the response set. We compared the locating results and the actual spatial positions of the animal to analyze the locating error of this network model. In Figure

Locating errors in three dimensional space of place cells.

Two sources account for the locating error. One is the systematic error of the network model. This is a model using a finite number of place fields to determine infinite even uncountable infinite number of locations in three dimensional space. This will cause inevitable error. And the degrees of freedom for spatial location is three, but after receiving three sensory inputs, place cell integrates this three dimensional information into one dimensional variable, which is firing power. Restoring the three dimensional information from one will clearly cause error. This is systematic error for the model. The other source of error is the inaccuracy of sensory. In order to simulate the inaccurate estimation of distance to landmark of the animal, we add the error term to the sensory model. This will also lead to the locating error. Excluding this term could reduce the locating error to a lower level. However, the final locating errors are limited under a certain boundary. So this network model can successfully perform the locating function in three dimensional space.

The total energy consumed by these 200 place cells during the exploration process is illustrated in Figure ^{6} nJ. And the minimum is close to zero. Many cells remain a low energy cost while preforming the locating function, this implies that during the spatial representation process, the neural system complies with the energy efficiency principle. It means to code the neural information with minimum energy consumption (Wang R. et al.,

Total energy consumption during exploration of place cells.

We have constructed an energy model of place cells to perform the locating function in three dimensional space. In this model, an important parameter is σ_{j}, as mentioned in section Model and Method, which is a random variable complying with Gaussian distribution, which influences the size of place fields (Kulvicius et al.,

Larger place fields and higher energy consumptions.

Meanwhile, higher energy consumption and larger place fields imply that during the spatial learning process, more place cells obtain highly overlapping place fields, place cells with similar initial weights may become more alike, resulting in the correlations of place fields become higher. This phenomenon is shown in Figure

Linear correlation of place field centers.

Another evidence can be seen in Figure ^{6} nJ energy, while the locating errors are larger than the small field situation (Figure

Energy consumed and locating errors. Total energy consumption

More simulations with a larger range of place field sizes suggest that the final locating error was not simply monotonously increasing as the place field enlarging. There always exists a minimum localization error when the place field is of the medium size (Figure

Mean locating errors with respect to place field sizes.

When place field is small, the total energy consumption is not normally distributed among 200 place cells whereas the normal distribution hypothesis is failed to be rejected in large-place-field situation (α = 0.01) (See Figure

Total energy consumption distribution of place fields. Histogram

The three dimensional spatial tuning of this network model is comparable with the valuable experiment recordings. Figure

Model result compared with experimental recording (Yartsev and Ulanovsky,

Energy efficiency is one of the most remarkable features of the neural systems. In mammalian brains, 1,000 trillion operations per second are carried out while only several watts of energy is consumed (Kandel et al.,

Energy consumption is positively related to the place field size. If the place field is too small, the locating system will fail to cover the local space and provide spatial information insufficiently. On the contrary, large place field will convey adequate even redundant spatial information at the cost of much more energy consumption. So the locating system has to balance the spatial coverage and energy consumption, which leads to a moderately medium place field size as this model shows. And the balanced field size and energy consumption jointly regulate a more accurate locating function. A coupling energy model of grid cell and place cell could be constructed in future study, which will help us understand the energy efficiency principle in medial entorhinal cortex-hippocampus circuit and further the whole spatial cognition system of the brain.

This is a preliminary model for three dimensional spatial representation system and certain factors are simplified. For example, it is known that animal rely on visual, auditory, olfactory, or somatosensory stimuli for orientation. While in this model, the sensory input to place cell network is in an abstract form without addressing the type of the cues. And the neural energy consumed by synaptic transmission is neglected in this model for simplification. These shortages will be interesting topics for future modeling study. However, this model, which emphasizes the three dimensional locating function and takes energy efficiency into consideration as well, may be the initial step to complete a comprehensive energy model for the brain's spatial representation system in realistic three dimensional world.

Aiming at improving the defects of the studies of place cells and energy coding, we constructed a place cell network model representing three dimensional space on an energy level. Then we defined the place field, place field center by energy. The spatial representation and locating functions of this model have been analyzed and the energy consumption properties related to place fields and locating accuracy have been studied. The computational results showed that the model successfully simulated the basic features of place cells. The spatial selectivity and sizes of place fields vary among individual place cells, and the locating error can be limited under an acceptable low level by choosing the reasonable parameters. Then we demonstrated the relationship between energy consumption, place field size, and locating error. Furthermore, we found that the minimum value of locating error will be obtained when the place field is of moderate small size. This may suggest that the place cell network balance the spatial coverage and the energy consumption to achieve an accurate locating function, which implied the energy efficiency feature of the neural systems. The simulation results matched with experimental data (Yartsev and Ulanovsky,

Besides the locating function, path-planning and navigation are also the crucial functions of the brain's spatial representation system. Other cells such as grid cell, border cell, and head-direction cell should be introduced in future studies. These models of different types of cells can also be generalized into three dimensional space by this similar energy method. Then the system error of locating may be reduced significantly and the model will acquire the path integrating and navigation function in three dimensional space. Moreover, whether the degree of freedom of the sensory input is higher in three dimensional space than on a plane remains to be testified by physiology experiments. If it can be verified that there is a certain group of cells in the brain responding solely to altitude information, the dimension of integrated signals can be extended and the locating accuracy can be improved in the model. These future works will help us understand and explain the three dimensional spatial representation system of the brain, and will further reveal how the energy efficiency principle would guide the brain to execute the locating, path planning and navigating functions. It will be a new view to study the mystery of the brain.

YW: co-designed the research, constructed the model, wrote software code, wrote paper; XX: co-designed the research, analyzed results, contributed to writing of software code and paper; RW: participated in research design, model construction and results analysis, edited paper.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.