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., 1998; Attwell and Laughlin, 2001; Crotty et al., 2006; Moujahid et al., 2011). Notably, energy is supplied to ion pump by the metabolism of adenosine triphosphate (ATP). The energy is primarily used to transport the ions against the ion concentration gradient. During electrical activity of a neuron, ions are driven by concentration gradient to cross the cell membrane and form the ion currents. The Joule heat due to resistances of ion channel conductance can be a convenient approach to understand neural energy based on an equivalent electrical circuit of neurons. The Hodgkin-Huxley (H-H) Model is the most successful model on the ion channel level. So the neural energy can be calculated by H-H model.

The equations of H-H model are:

where C_m is membrane capacitance of a neuron, V_m is membrane potential, E_Na and E_K are Nernst potentials of Na⁺ and K⁺, and E_l is the potential while there is no leakage current. g_l, g_Na, and g_K are, respectively, the leakage conductance, Na⁺ channel conductance, and K⁺ channel conductance. The typical values of these parameters are: resting membrane potential V_r = 67.3 mV, maximum Na⁺ conductance g_Na = 120 mS/cm², maximum K⁺ conductance g_K = 36 mS/cm², leakage conductance g_l = 0.3 mS/cm², 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., 1998; Attwell and Laughlin, 2001; Crotty et al., 2006; Moujahid et al., 2011; Wang et al., 2017a),

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., 2017a). This value, which can transfer the number of spikes into energy consumption, will be embedded into the network model later to determine the power and energy consumed by the place cell network. Then we can analyze the energy properties of the place cell network during three dimensional space exploration and localization.

We set up a cube space with a side length of L (see Figure 1). A bat or bird is placed randomly in a position of the environment, setting a task of learning the environment in the experiment. In this cube space, six borders are regarded as landmarks. The animal perceives environmental cues by using its visual (bird) or auditory (bat) system, then learns the sensory information by its neural network made of place cells.

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, 1996; Ollington and Vamplew, 2004) chose the distances to the four walls (East, West, North, and South) as the input to the sensory. We use the similar method but generalize it in the three dimensional space. Meanwhile, since only three variables among the six distances from the borders are independent, we choose the distances from L, F, and D walls as the independent inputs to the sensory.

Note that the actual location of the animal at time t is described by vector X(t) = (x₁(t), x₂(t), x₃(t)), where x_i(t) is the actual distance from wall i(i = 1, 2, 3) corresponding to L, F, D (Figure 1). However, it is important and reasonable to assume that sensory neurons of the animal are unable to acquire accurate perception of its own locations. So the perception input model is given by the following equation:

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., 2016).

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., 2000; Kulvicius et al., 2008). So the firing of place cells can be seen as the sum of inputs received from boundary vector cells (Kulvicius et al., 2008). And these boundary vector cells perform the sensory function. Then a feed-forward network can be applied to construct a model to describe the locating system constituted of place cells and sensor neurons, similar as the studies in two dimensional space (O'Keefe and Burgess, 1996; Hartley et al., 2000; Kulvicius et al., 2008; Yan et al., 2016). Sensory neurons input to place cells with X (t)′. This is a fully-connected network between layers where every sensory neuron is connected to every place cell with weights matrix W (t) = [w_ij(t)]_3×N, where i = 1, 2, 3 represent three sensory neurons, j = 1, 2, …, N represent the N place cells. Three sensory neurons perceive the distances from boundary L, F, and D. W (t) is a 3 × N matrix, and w_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., 2008),

Where γ is a random number uniformly distributed on the interval [0, 1], that is γ ~ U (0, 1). E(γ) which equals to 0.5 is the expectation of the uniformly distributed random number γ. The weights distribution is shown in Figure 2 after initialization (Kulvicius et al., 2008).

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., 2008). Weights are the basis vectors in the model, which are used to compute firing powers of place cells. When competitive learning rule is employed, place cells become tuned to a specific input, which leads to the spatial selectivity.

Inspired by the firing rate model in two dimensional space proposed by O'Keefe and Burgess (1996) and Hartley et al. (2000), we construct the following model which combines the neural energy method to represent three dimensional space as follows,

Where P_j(t) is the firing power of the jth place cell at time t, C is the energy consumption by a place cell during an action potential. As introduced earlier, ~188 nJ energy is consumed to transmit a spike. In order to reflect the diversity of place cells' metabolic environment, C is normally distributed from N (188, 10) nJ. R_m is the maximum firing rate of a single place cell, which is about 20 Hz (Hartley et al., 2000). n is the number of sensory inputs and W_j(t) is the jth row of W(t). The norm is the Euclidean distance. And σ_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., 2008). Obviously, efficiency of this mechanism is low because only the weights of one winning cell are changed at every step. Meanwhile, the firing rate learning model is inconvenient to generalize to multiple levels. Therefore, learning rule is modified not only in a batch manner (Yan et al., 2016) but also on an energy level as follows,

Where μ is the learning rate, P_thr is the responding threshold represents the minimum firing power of cells that are activated. J is the response set. And every place cell responding to the current location with a firing power above threshold will modify the weights from sensory neurons.

The firing power of place cell j can also be viewed as the function of spatial location X(t)′ according to Equation (5). Then the place field of cell j can be defined as the set of all the location X(t)′ with firing power larger than P_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 j is defined during this dynamical process as follows,

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

Where, Loc(t) is the location of the animal at moment t determined by this locating model. And C_j is the place field center of cell j, while P_j(t) is the activity power of cell j at moment t.

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 3 shows the electric power of each ion channel during an action potential of a place cell. Green line is the total power and red, black, yellow, and fuchsia lines are powers of Na⁺, K⁺, leakage, and stimulus currents, respectively. By integrating the power over time, we can get that one action potential costs about 188 nJ energy.

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 4. The coordinates of three dimensions are labeled as x, y, and z. The highly random trajectory covered most of the cube space. This behavior is reasonable for an animal in the absence of food or water reward. Due to the different initial weights, the preliminary responses of the place cells are not the same. Then according to the batch competitive learning rule, each place cell acquires unique spatial selectivity, and form its own place field.

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

Figure 5 displays the activity patterns of 16 randomly selected cells. The scatter plots in space represent the different locations of the animal in the random search trajectory, and the colors indicate the firing power of place cells at the corresponding position. We can further know if a certain location belongs to the place field of a cell, that is, whether the cell has spatial selectivity for this location. It can be seen from Figure 5, due to individual differences of place cells, the distribution and size of place fields as well as firing powers vary among different place cells. These facts can further be obtained from the histograms in Figure 6 which illustrates the distributions of maximum firing power (left) as well as the size of place field (right) among the 200 place cells. The vertical axes both represent number of neurons and the horizontal axes represent maximum firing power and size of place field respectively. Note that the size of place field are reflected by the count of locations at which the cellular activity is above threshold. Maximum power is about 3,000 nW among these 16 random selected cells in Figure 5. Larger place fields usually have higher maximal power. This may be the consequence of competitive learning rule.

Normally the place field centers are near the locations with maximum powers. Two hundred place field centers are summarized in Figure 7. As can be seen from the figure, the 200 field centers are scattered throughout the space. The phenomenon mentioned by Kulvicius et al. (2008) that place field centers may concentrate near the spatial center is not occurred. The density of place field centers is quite uniform. Plenty of place field centers are near the space borders.

Figure 8 shows the energy place fields of four randomly selected place cells. From this figure, the individual differences and the unique spatial selectivity are clearly revealed. Place cell a has a smaller place field near wall B, R, and U. Place field of cell b is larger, which is one of the neurons with higher energy consumptions.

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 9A, we chose the last 100 steps during the random exploration and calculated the locating errors for these 100 locations. As the figure shows, the error between locating and actual position fluctuates under an acceptable low level. Errors of all 10,000 steps during the exploration are shown in Figure 9B.

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 10. The horizontal axis is number of place cells, and vertical axis is the total energy consumption. The maximum energy consumed by a single cell is close to 1.7 × 10⁶ 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., 2015).

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., 2008). The expectation of this distribution affects the mean size of place fields, and the standard deviation affects the variability of different place fields. Figure 11 depicts two groups of place fields with greater size. As showed in the figure, the enlargement the place fields can expand the spatial range of high power activity of place cells (Figure 11B). Notably, the field size and selectivity will still evolve dynamically as the exploration and learning proceed. While the place fields are too large, place fields belong to different cells will have more overlapped part. During coding a position in space, several place cells will be activated at the same time and are more likely to response with higher powers. This is not economical or reasonable from the energy perspective. So the spatial representation system of the brain is possible the balancing result between spatial coverage (accuracy) and energy efficiency.

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 12. Two hundred field centers show a strong linear correlation, and concentrate in the center of space. This suggests that in the case of high power consumption, place cells have a high redundancy coding the spatial information and consume too much unnecessary energy. This result once again confirms the economic principle of the energy usage in the information coding of the brain.

Another evidence can be seen in Figure 13. Total energy consumed by 200 place cells with larger place fields (Figure 13B) and the corresponding locating errors (Figure 13D) are shown in this figure. Unlike Figure 13A, all the cells consumed more than 4 × 10⁶ nJ energy, while the locating errors are larger than the small field situation (Figure 13C). This suggests that the energy consumption in the neural system is not the more, the better.

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 14). So the place field with a reasonable optimal size will most accurately preform the localization function.

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 15). Energy of larger place field cells is closer to normal distribution. Since information can be seen as the degree of unexpectedness, it will not be totally random. So the difference between neural energy distribution and the normal distribution may be crucial to understand neural information coding. And cells with moderate smaller place fields consumed less energy possibly contain more spatial information (Brown and Backer, 2006).

The three dimensional spatial tuning of this network model is comparable with the valuable experiment recordings. Figure 16 illustrate the comparison between model result and experiment data recorded from bat (Yartsev and Ulanovsky, 2013). Figure 16A shows the spikes (red dots) overlaid on bat's position (gray lines), and Figure 16B is the three dimensional color-coded rate map, with peak firing rate of 15 Hz. Figure 16C is the model simulation of the typical activity pattern of a place cell. By comparing these figures we can find out that the simulation result is similar with the experimental result morphologically. In this point of view, the behavior of this model is matched with the experimental data.

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.