Neuromorphic place cells

Part of the model that is implemented is an improvement upon the original OI model, tailored for a more robust and simpler implementation in silicon. It can compensate for the inevitable behavioral variations of either biological neurons or neuromorphic implementations while reducing the interference operation to logic operations. The performance and details of the model are published in [22], though we will introduce its basic structure here for completeness. An important characteristic is that to accommodate variations among theta cells readily, the cell formed by the improved model is downgraded into an egocentric place cell reporting segmented displacement rather than being a location-specific cell in the original OI model. In this section, we first review how to form a place cell in the ideal scenario, then discuss why we can get only an egocentric place cell from either the original OI or the improved model. Later in section 5, we will discuss how to integrate the segmented displacement into localization with respect to a global reference frame.

As predicted by the OI models [16, 17], the fundamental components of an OI model are oscillators that can encode the animal's moving velocity into frequencies. Then the integration of velocity can be translated into the accumulation of angular velocities, thus achieving path integration for localization. This type of neurons was later observed and reported in [19] with their oscillatory behavior is related to the animals' traversing velocity as described in equation (1)

$$\begin{equation}\begin{array}{*{20}{c}} {F = {F_{{\text{idle}}}} + \beta \left( {\overset{{\scriptscriptstyle\rightharpoonup}} {V} \cdot \overset{{\scriptscriptstyle\rightharpoonup}} {V}_{{\text{p}}} } \right)} \end{array}.\end{equation} \tag{ 1 }$$

Here, $F$ is the oscillation frequency of a theta cell. It is centered at a constant idling frequency ${F_{{\text{idle}}}}$ and modified by a scaled inner product between its preferred direction unit vector $\overset{{\scriptscriptstyle\rightharpoonup}} {V}_{{\text{p}}}$ and the agent's velocity $\overset{{\scriptscriptstyle\rightharpoonup}} {V}$. The scaling coefficient $\beta$ is the gain or frequency response factor. The parameters $\beta$ and ${F_{{\text{idle}}}}$ are relatively stable for one theta cell but differ between theta cells as Welday et al discovered [19].

In the OI models, grid cells and place cells can be obtained by interfering with various theta cells [16, 17]. First, spatially periodic bands or gratings can be formed by interfering a theta cell with a cell that has an oscillation frequency at ${F_{{\text{idle}}}}$, with the results shown in subplots (a)–(c) of figure 1. The process resembles the down conversion from the demodulation of wireless communication to remove the 'carrier frequency' ${F_{{\text{idle}}}}$ then extract the $\beta ( {\overset{{\scriptscriptstyle\rightharpoonup}} {V} \cdot \overset{{\scriptscriptstyle\rightharpoonup}} {V}_{{\text{p}}} } )$ term that encodes spatial information. The resulting signal is the cosine of the phase accumulated by the moving velocity projected onto the preferred velocity of the theta cell. Thus, its spatial firing pattern forms a periodic sinusoid along its preferred direction. The difference in periodicity of subplots (a)–(c) in figure 1 is due to distinct scaling coefficient $\beta$, or the magnitude of the theta cell's preferred velocity vector. Grid cells can be formed by interference between gratings of preferred direction that are multiples of 60° apart, as indicated by the red dotted arrows in figure 1. Place cells can be formed in a similar fashion but with gratings with a much larger period, which can be obtained through interference between denser gratings but with mutually prime scaling coefficients, as shown by the green arrows in figure 1.

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The improved OI model employs a similar interference structure in general but discovered several realistic factors that lead to the improvements and alternation from place cells to egocentric place cells. In an analog implementation or a biological neural circuitry of a theta cell, it is unlikely that all theta cells share the same baseline frequency ${F_{{\text{idle}}}}$. Even with a distributed pairing strategy, the chance of finding two theta cells with the exact same ${F_{{\text{idle}}}}$ is minimal. Here in equation (2), we denote the term ${F_{{\text{of}}{{\text{f}}_n}}}$ as the offset frequency for the nth theta cell with respect to an average baseline frequency among all theta cells

$$\begin{equation}\begin{array}{*{20}{c}} {{F_n} = {F_{{\text{idl}}{{\text{e}}_n}}} + {\beta _n}\left( {\overset{{\scriptscriptstyle\rightharpoonup}} {V} \cdot \overset{{\scriptscriptstyle\rightharpoonup}} {V}_{{\text{p}}n} } \right) = {F_{{\text{idle}}}} + {F_{{\text{of}}{{\text{f}}_n}}} + \beta \left( {\overset{{\scriptscriptstyle\rightharpoonup}} {V} \cdot \overset{{\scriptscriptstyle\rightharpoonup}} {V}_{{\text{p}}n} } \right)} \end{array}.\end{equation} \tag{ 2 }$$

In the improved model, with schematics on the bottom of figure 2, an offset-reduction strategy is proposed to reduce the impact of this offset term by interfering between theta cells that have similar ${F_{{\text{off}}}}$ but opposite preferred directions. We model the logic AND interference operation as multiplication here with operation symbol ⊗ to indicate that only the low frequency component is considered. The following equation demonstrates the effect of the interference in the signal domain. The actual oscillating signal of any theta cell contains an initial phase ${\varphi _i}$, which is critical to determine the location of the place field in the OI model for place cells

$$\begin{align}& \cos \left( {2\pi {F_a}t + {\varphi _a}} \right) \otimes \cos \left( {2\pi {F_b}t + {\varphi _b}} \right) \nonumber\\ & \qquad\qquad = \cos \left( {2\pi \left[ {\left( {{\beta _a}\left( {\overset{{\scriptscriptstyle\rightharpoonup}} {V} \cdot \overset{{\scriptscriptstyle\rightharpoonup}} {V}_{{\text{p}}a} } \right) + {F_{{\text{idle}}}} + {F_{{\text{of}}{{\text{f}}_a}}}} \right) - \left( {{\beta _b}\left( {\overset{{\scriptscriptstyle\rightharpoonup}} {V} \cdot - \overset{{\scriptscriptstyle\rightharpoonup}} {V}_{{\text{p}}a} } \right) + {F_{{\text{idle}}}} + {F_{{\text{of}}{{\text{f}}_b}}}} \right)} \right]t + \left( {{\varphi _a} - {\varphi _b}} \right)} \right) \nonumber\\ & \qquad\qquad = \cos \left( {2\pi \left[ {\left( {{\beta _a} + {\beta _b}} \right)\left( {\overset{{\scriptscriptstyle\rightharpoonup}} {V} \cdot \overset{{\scriptscriptstyle\rightharpoonup}} {V}_{{\text{p}}} } \right) + \left( {{F_{{\text{of}}{{\text{f}}_a}}} - {F_{{\text{of}}{{\text{f}}_b}}}} \right)} \right]t + \left( {{\varphi _a} - {\varphi _b}} \right)} \right)\,.\end{align} \tag{ 3 }$$

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This operation suppresses the phase error accumulation. Notice that the result has a very similar mathematical form and spatial pattern as the original theta cell, with a new frequency response factor $\beta = \left( {{\beta _a} + {\beta _b}} \right)$ and a new offset frequency ${F_{{\text{off}}}} = {F_{{\text{of}}{{\text{f}}_a}}} - {F_{{\text{of}}{{\text{f}}_b}}}$. Furthermore, the summation of frequency response factors $\left( {{\beta _a} + {\beta _b}} \right)$ makes the $\vec Vt$-dependent term in (3) more dominant than the subtracted offset frequency, with the effect shown in subplots (a)–(c) of figure 2. Because of the similarity in behavior, we can view the result as an effective theta cell. This modularity of the offset reduction strategy enables a layered structure as shown in the schematic in figure 2, should the variation be large. This strategy also eliminates the need for idling oscillators and consists purely of theta cells, further reducing system complexity. The variation in $\beta$ among theta cells is less of a problem since it contributes to the uniqueness of the egocentric place's firing field, similar to the method of spacing by mutual primes discussed previously. The uniqueness problem is a more predominant issue due to our simplification that converts the sinusoidal oscillation profile of theta cells to square-wave since it basically regularizes the positive part of a sinusoid to one. Please refer to the original publication [22] for a more detailed discussion. In conclusion, the variations in both $\beta$ and ${F_{{\text{off}}}}$ becomes advantages when forming a unique firing field for place cell while applying a square-wave oscillation profile, enabling us to use logic AND operation for interference to save resources.

In the actual scenario however, we understand that the offset frequency ${F_{{\text{off}}}}$ cannot be eliminated all the way to zero, so the phase shift computation must be adjusted as in (4) due to the remaining offset frequency, even after offset reduction. The modulo operation is to account for digitized phase shift in an implemented system, such as a ring oscillator described in the next section that offers 8 digitized phase shifts, or a neural circuit model of theta cell in the form of a ring oscillator [20, 23]

$$\begin{equation}\begin{array}{*{20}{c}} {{\varphi _i}\left( {R,{ }\theta } \right) = \bmod \left( {1 - {\text{mod}}\left( {R\left( {\cos \left( {\theta - {\theta _{{\text{p}}i}}} \right){\beta _i} + \frac{{{F_{{\text{of}}{{\text{f}}_i}}}}}{{| {\overset{{\scriptscriptstyle\rightharpoonup}} {V} } |}}} \right),1} \right),{ }0.125} \right)} \end{array}.\end{equation} \tag{ 4 }$$

Here, ${\varphi _i}$ is the phase shift of the $i$th effective theta cell after our proposed AND interference for the egocentric place cell designated for location $\overset{{\scriptscriptstyle\rightharpoonup}} {x}$. $R$ and θ are the polar coordinates for $\overset{{\scriptscriptstyle\rightharpoonup}} {x}$, and ${\theta _{{\text{p}}i}}$ for ${\overrightarrow {{V_{\text{p}}}} _i}$. The spatial firing patterns of two egocentric place cells with arbitrary designated locations are shown in figures 2(d) and (e). The modulo 0.125 operation is to accommodate the eight-phase digitized output of the theta chip, which will be introduced in the next section. They are formed from the same group of 48 simulated theta cells with a 25% relative standard deviation on idling frequency and a 20% relative standard deviation of frequency response factor.

The reason for the degradation from place cell to egocentric place cell is because of the remaining ${F_{{\text{of}}{{\text{f}}_i}}}$ term. They only operate when the theta cells are all in a known state and the agent is traveling with a constant velocity when reaching any point in the spatial maps of figure 2, due to the phase accumulated through ${F_{{\text{off}}}}t$. A phase reset procedure is proposed in [22] to address this problem. The procedure is triggered either 1: at a fixed interval when velocity is constant or 2: in the event of changing the movement velocity. The first condition is to clear unforeseen errors, such as thermal noise or crosstalk, accumulated by regularly pulling the system back to a known state. The second condition is due to the dependence of phase accumulated with time induced by the remaining offset frequency, generating a path dependency from the origin to any point. For example, the agent can travel from the origin to [1,1] directly, or by detouring via [1,0]. Though both paths end at the same location, the difference in travel times leads to different phase accumulations across the theta cells. Thus, each segment of the path must be recorded as a displacement vector, triggering the phase reset event and creating a new egocentric frame at the point of velocity change. Every time the phase reset event happens, the system can record the current egocentric place cell firing to achieve a path integration or, in other words, a dead reckoning. In section 5 of this paper, we describe our proposed model for place cells to perform localization through the accumulation of egocentric place cell activities.

A theta chip has been designed and taped out with TSMC 65 nm technology, targeted at producing the behavior described by equation (1). Though a neural attractor network model was previously proposed in [20], it consists of many neurons with both inhibitory and excitatory interconnections, making VLSI implementation and configuration difficult. In order to provide an array of programmable theta cells with reasonable hardware resources, we choose to implement the theta chip with abstract neomorphism by reproducing the theta cells' behavior model. But, to preserve an analogy to the neuronal behavior, we employ a mixed-mode design with analog computation circuitry to reduce transistor count as well as a continuous-spectrum response to the input velocity. We also choose digital output for the theta unit oscillations for easier handling by external components. We briefly describe the final design of the chip below; a detailed description of an early design of the included circuits is discussed previously in [21].

The chip consists of 128 theta cell units with an I/O (input-output) arbiter for the control, program, and output of the theta units. Each theta unit has an individually programmable preferred velocity and operates asynchronously. A theta cell unit includes four major components.

• (1)  
  SRAMs: Each unit has two 4-bit SRAMs to store the x and y components of the preferred velocity $\overset{{\scriptscriptstyle\rightharpoonup}} {V}_{\text{p}}$. The values are signed and in ascending order with value 8 (binary: 1000) representing the zero value. They can be programmed during the chip's start-up phase.
• (2)  
  A/D Converters: Since the preferred velocity vectors are not to be changed after configuration, capacitance DACs that require refreshment are not preferred here. We adopted a W-2W transistor ladder [24] with a long transistor length to minimize current consumption. Two such DACs are implemented for each theta unit, connecting to the preferred velocity SRAMs.
• (3)  
  Analog Computation Module: The analog computation unit calculates the inner product between the theta unit's preferred velocity and the input movement velocity. We use two Gilbert cells to compute a continuous four-quadrant dot product between the internal preferred velocity and the movement velocity input to the theta unit, both represented in analog voltages. The Gilbert cells require less power and fewer transistors than conventional multipliers. The analog multiplication generates a pair of differential voltages as the product. Two pairs of differential voltages, representing the products of the x and y components, are converted into two currents using differential pairs, and the sum of the two currents represents the dot product, as shown in figure 3(c).
• (4)  
  Oscillator: The basic structure of the oscillator is a current-starved ring oscillator as shown in figure 3(b), where the control current for the ring oscillator is mirrored from the dot product circuit. An extra NMOS transistor is added between the inverters to allow a manual hold for the oscillation with the control signal Cap_clear, the model needs this phase reset capability for pulling the oscillation to a known phase for correcting accumulated error. To achieve a linear relation between the current and frequency, as well as to reduce the oscillation frequency, extra load capacitance is introduced between inverter stages. Eight of the nine inverters are connected in parallel with a delay capacitor taking advantage of the Miller Effect, thus ultimately reducing the capacitances needed down to 1 pF each. This design provides a digitized eight-phase output of each theta cell unit for the improved OI model to choose from through equation (4). And, since the TSMC technology provides metal-metal capacitors in upper layer metals, the capacitors are placed above the transistors to reduce the area needed for the entire design.

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The I/O arbiter follows the concept of Time Division Multiple Access for the I/O of each theta unit, according to the schematic shown in figure 3(a). It comprises a chain of 1024-bit shift registers that forms 128 groups of 8-bit registers. Within the group, each bit enables one phase output of a theta unit, and the first bit also provides an enable signal for programming the preferred velocity SRAM of this unit. Each register is accompanied by a bypass SRAM that determines whether the shift path goes through the register or bypasses it. The bypass capability allows a shorter scan cycle after the calibration and can only provide signals that are useful for further processing. Upon start-up, the shift register will shift through all the theta units and their 8 phases to program each theta unit's preferred velocity and the Bypass SRAM for each phase output.

The chip is placed on a custom PCB to interface its digital ports with the FPGA and to meet its analog bias requirement by a DAC. The chip employs serial input and output. Preferred velocity input is transmitted over an 8-bit bus with the lower 4 bits carrying the x component value and the upper 4 bits for y. Upon startup, the clear signal needs to be held high for a few clock cycles to guarantee reset of all SRAMs and shift registers. The first bit of the long shift register will be held high to index the first theta unit's preferred velocity SRAM and the Bypass SRAM for the first phase of the oscillation output of this unit. Then if pulling the active high enabling signal Write, the data input signal bypass and the preferred velocity byte can then be used to program the theta unit's output and preferred velocity. Each clock pulse proceeds the shift register to program whether to output each phase of an oscillation, and every eight clock pulses will move the index to the next theta unit, eventually looping through all 128 units. After the initial setup, the clock then only scans through the phases that are programmed to be output, reducing the length of the scanning cycle or, equivalently, increasing the sampling frequency without increasing the clock frequency. The user can then load the moving velocity of the agent either digitally through the preferred velocity byte by pulling load signal high, or through analog voltage using two analog input pins.

As suggested by the improved OI model, the first layer of interference should be between theta units with similar ${F_{{\text{idle}}}}$ but with opposing preferred directions for offset reduction. The total number of layers needed depends on the distribution of ${F_{{\text{idle}}}}$. In this demonstration, we implemented only two layers to show the capability of our model with a small number of operations. In figure 8 in the results section, 82 of the units exhibit a coefficient of determination higher than 0.9. Thus, the structure of a two-layered network will have 40 interference pairs in the first layer and 20 pairs in the second. Among the participating 80 units, units are paired to have the closest possible idling frequencies. The output node will be the direct logic AND operation on the outputs of the 20 nodes of the second layer, with the structure shown in figure 4.

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Since the velocity is represented in Cartesian coordinates, the straightforward choice of preferred directions for the theta cells are positive and negative directions along the x and y axes. Given the programmability of the theta units, we can first form 40 pairs based solely on the closeness of ${F_{{\text{idle}}}}$, then assign 20 pairs with opposing velocities along the x axis, and the other 20 along the y axis. In each pair, for example, one theta unit might be programmed to have a preferred velocity of [4, 0] and the other with [−4, 0]. In the second layer, we can choose to interfere between one from the x-aligned group and one from the y-aligned group. This operation resembles the formation of grid cells as shown in figure 1(i-iii), showing that one potential functionality of the grid cells is to reduce the influence of offset frequency. An alternative is to repeat the process for pairing based on the similarity of frequencies. Thus, the network consists of 60 interference operations, each of which comprises a logic AND between two signals and a low-pass filtering process as illustrated in figure 4, which is an instantiation of figure 2 but with an inherent formation of grid cells.

The network for the egocentric cells is implemented with the Xilinx Spartan-6 FPGA. As shown in figure 4, the implementation comprises two main components: an interference node and a multiplexer. Each node contains simply an AND gate followed by a filter to remove the high-frequency component as suggested by the models. In systems like biological neural networks or similar neuromorphic network systems such as IFAT [25–27], integrators like the neuron's cell membrane or a capacitor for an integrate-and-fire neuron model can perform this low-pass filtering. In our current implementation in the FPGA, we mimic such functionality through a 9-tap Hamming window function followed by a digitized RC circuit. And, to show the compatibility of the system's filters, the filters in the second layer of nodes use a 9-tap moving average filter with a digitized RC circuit with a larger time constant. All the filters are arranged in a pipeline to ensure a continuous data flow and to allow the network output to be synchronized with the clock. The output node, which is simply a 20-input AND gate, produces the output of the place cell which is then buffered in a FIFO queue for streaming to a PC for the next layer, as will be described below.

The multiplexer in figure 4 interfaces between the theta chip and the network described above. For efficiency, the data path structure between network layers is fixed, thus localizing the job of network configuration to the first layer's input. It scans a cycle of the theta chip and directs each oscillation output to its position in an 80-bit wide buffer based on a lookup table, whose values are computed from equation (4) with the frequency information obtained from a calibration cycle. After the multiplexer finishes each scan cycle, it generates a clock signal for the network to load in the 80-bit buffer and advance the interference process.

As described above in section 3.2, the network in the FPGA needs to initialize the theta chip before operation. Based on equation (4), a place cell's designated location is determined by the initial phase shift relation among the theta cells. Combining that outcome with equation (3), every interference result is the difference between the initial phases of the source signals. Thus, in our 2-layered network, the output of the second layer depends on the initial phase of 4 signals. If we initialize 3 of the signals with the same initial phase, the difference between the initial phase of the fourth theta cell with the other 3 can determine the initial phase of the node of the second layer. So, the number of phases to be output by the theta chip is programmed to be $60 + 8 \times 20 = 220$, where one of every 4 theta units has all 8 phases available for the multiplexer, denoted as the 8 dots around one theta unit in figure 4. After the theta chip is initialized, we gather the traces for the 80 theta units and compute their frequency to compute the phase connection lookup table, given the desired location of a egocentric place. In this form, any egocentric place (allowing for digitization error) within the range can be generated with the same basis of 220 theta phases by a different lookup table for the multiplexer. Their results for tracking the movement within a certain range are demonstrated in figure 10 in the results.

After we received the theta chip from TSMC then placed on a testing PCB to interface with the FPGA asshown in figure 6, We first evaluate the theta chip for its accuracy in reproducing the oscillation frequency behavior described in equation (1). To have a more thorough evaluation of the performance, we supplied analog voltages that override the internal DACs for digital inputs. As a few units' responses shown in figure 7, the frequency demonstrates a sigmoid relationship with the inner product value. Thus, during the operation, we chose to limit the magnitude of velocities to approximate the linear relationship. The limitation on the range lead to a shrink in frequency swing as well, allowing us to tune down the current supply for the oscillators without concerning them reach equilibrium on deep negative inner product values. This shifts the idling frequency lower from around 6500 Hz–2000 Hz, reducing the power consumption of the chip during actual operation.

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The characteristics of the theta chip configured for operation are shown in figure 8. Figure 8(a) depicts a theta cell's selectivity on the input velocity direction. To assess the performance of the theta chip, we exhaustively program all combinations between the preferred velocity and the input velocity for every theta unit, then record the traces. Frequency analysis then gives a relationship between the inner product of the input and preferred velocities with the oscillation frequencies, with a selection of relevant results with actual configuration values used for egocentric place construction are shown in figure 8(b). In general, most of the theta units exhibit a positive monotonic relation between the inner product value and the oscillation frequency. Figure 8 shows a scenario in which the input velocity's x and y components are scanned separately while the preferred velocity is clamped to zero; it shows that some of the variation comes from the W-2W DAC. A slight offset between the zero value of the DAC output and the broadcasting zero-value reference voltage for the mixer resulted in non-zero multiplication results added to the resultant frequency.

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Given the relationship between preferred velocities, input velocities, and oscillation frequencies of the theta units, we fit the data for each theta chip with equation (1) to evaluate their performances. The results, shown in figure 8, demonstrate variations between the units. Later in the implementation, we choose to only use theta units with a correlation of R² > 0.9. As shown by figure 8(d), the values of $\beta$ and ${F_{{\text{idle}}}}$ vary considerably among the theta units due to the analog implementation of the computation unit and the oscillator. As described in section 2.2 and in [22], the variation of $\beta$ is in fact beneficial in our model since it substitutes the need to deliberately program parallel preferred velocities to be mutually prime. This variation also avoids the need for a very large frequency domain of operation since the magnitude of programmed preferred velocity can be the same across the units, which helps the oscillators to operate without saturating the current starving capabilities of the ring oscillators. Furthermore, the non-uniformity better resembles the biological behavior of neurons and can, therefore, better test the robustness and plausibility of the proposed improved OI model.

We demonstrate the output flexibility due to the bypass functionality of the chip in figure 9. Although the chip's output is serial, by knowing the number of phases programmed to be output, we could re-parallelize the data stream to reconstruct the oscillation of each outputting phase. Figure 9 demonstrates 8 consecutive phases programmed to be output: the first 4 traces belong to different phases of the same theta unit, and the last 4 traces are from different theta units with different preferred velocities. Based on the measurement, an oscillation frequency range between 1500 Hz and 3000 Hz can produce reasonable linearity with the lowest frequency range. Thus, the Nyquist frequency of the scanning clock becomes $\left( {2 \times 4000 \times N} \right)$ Hz, where $N$ is the number of outputting phases. In the experiments, we chose a sampling clock of 6 MHz with 220 outputting phases from the theta chip, resulting in a 27 272.7 Hz sampling frequency for each phase.

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To ensure the formation of egocentric place cells, phase accuracy is crucial. In figure 9, we plot the recorded output of a theta unit's 8 phases and the reconstructed signal with the frequency computed from the Fourier transform, and the respective phase shifts digitized at 0.25 π steps. Most of the units show a good correlation between prediction and output. Adaptive algorithms are also applied when configuring the egocentric place networks to ensure the closest fit between the phase tap and the computed phase ${\varphi _i}$. For cases when the resulted nodes cannot provide constructive interference, eventually, due to cumulative phase alignment error among the participating theta cells, the egocentric place network will simply ignore this group.

An instance of the egocentric place network is implemented in the FPGA for validation. We want to test the capability of the egocentric place cells for tracking movements within the domain and the state of aliasing among all the potential egocentric place cells. We generated a grid of 11 × 11 egocentric place cells as the domain, but due to resource limitations on the FPGA, we could only conduct the experiments for each egocentric place within the domain under the same input when building a whole map for the response of all the cells. In other words, the lookup table for each egocentric place is reprogrammed for the same input velocity. The start of each session is signified by the release of the Cap_Clear signal of the theta chip, then we record the network output. A selection of traces is shown in figure 10, where each square shows whether the value of its corresponding egocentric place is high or low. Here, we show snapshots of two trails of the egocentric place cells firing sequentially as the input velocity is held constant. Each egocentric place cell activates only when the agent reaches its place field; this demonstrates its localization functionality. If several of them are deployed simultaneously in combination with the place cell model proposed in section 5, real-time localization can be achieved. Though not shown in the figures, we also observed artifacts from other egocentric place cells coincidentally briefly firing. A similar phenomenon can also be observed in the traces of the relevant egocentric place cells in figure 10, but they are not significant enough to generate a pulse in the subsequent layer since their pulse width are significantly shorter and contains much less energy. Only pulses significant enough would trigger an activity bump migration in the place cell's layer, and thus will not affect the tracking accuracy of the place cells. We have tested various types of window functions with different numbers of orders. Hamming windows with orders larger than 15 can provide egocentric place cell signals with much fewer artifacts. But considering the balance between performance and resource demands for hardware implementation, a 9 tap square window function is chosen. In the experiment shown at the bottom of figure 10, more artifacts can be observed for egocentric place cells with place field further from origin. This is due to the accumulation of unpredictable thermal noise from the theta chip over a longer time. The phase reset mechanism described earlier is intended to restrict the magnitude of noise accumulation and prevent it from getting enough energy to disrupt the tracking in the place cell layer.

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The location tracking mechanism is powered by the basis formed by the egocentric place cells. Though we discussed the potential advantages of an abundant egocentric place basis, the limitation of the FPGA, especially the need for simulated RC filters on board, limits the number of egocentric place networks that can be implemented. As shown in figure 4, implementing just one network requires 80 nodes that each involve a 9-tap FIR and a multiplier for the simulated RC filter. But, since the input is digitized and the FIR is a square window function, the FIR only requires counters. We expect all these excessive components can be easily replaced by any existing programmable neuromorphic device that has membrane capacitance imitations such as IFAT, or a mixed-mode ASIC chip, after the system is finalized.

We chose to implement the conventional Cartesian basis on the cardinal directions (N, S, W, E), so 4 such networks need to be instantiated. However, on close observation, due to our 2-layered structure requiring only 1 out of 4 theta cells to have a programmable phase, half of the first layer's nodes are constant and thus can be shared with other vector networks to save resources. In general, the number of sharable nodes is:

$$\begin{equation} S = \mathop \sum \limits_{n = 1}^{N - 1} \frac{{{2^n} - 1}}{{{2^N}}}M = M\left( {1 - \frac{{N + 1}}{{{2^N}}}} \right).\end{equation} \tag{ 5 }$$

Here, $S$ is the number of sharable nodes, $N$ is the number of layers and $M$ is the number of theta cells involved. Thus, the total number of nodes $Q$ for $K$ egocentric place networks is

$$\begin{equation} Q = M\left( {1 - \frac{{N + 1}}{{{2^N}}}} \right) + KM\frac{N}{{{2^N}}} = M\left( {1 + \frac{{\left( {K - 1} \right)N - 1}}{{{2^N}}}} \right) .\end{equation} \tag{ 6 }$$

The number of nodes or resources needed for multiple networks becomes less significant as the number of layers increases. In the current setup, we could use $20 + 4 \times 40 = 180$ nodes instead of $4 \times 60\, = \,240$ for the implemented basis, avert needing more DSP capacity than is provided by the SPARTAN-6 FPGA used here.

To conduct a tracking experiment, commands for controlling input velocity and resetting to the theta chip are issued from the host PC to the FPGA then relayed to the theta chip, then the PC records the output from the egocentric place cells and feeds the traces into the place cell network described in section 5. The Cap-clear signal is generated at the beginning of the trail and when any egocentric place cell fires, or a change in input velocity. Currently, for ease of display, the place cell network is simulated in MATLAB. It is trivial to implement the place cell network in its current state on a FPGA or a general-purpose neuromorphic chip given the modular design and simple logic. But we believe it can be further developed to perform mapping and navigation functionalities with the interconnection between place cells; this prospect will be explored in the discussion.

Three of the experiments in figure 11 demonstrate tracking capability with the implemented egocentric place cell networks receiving inputs from the theta chip. The travel is sequenced into segments as shown by the color code in figure 11. In the current experiments, each segment is encoded as a change of velocity command to load the theta chip's input DAC with the new velocity and held the Cap-clear signal high to commence a phase reset. The first path simulates the agent traveling along while randomly looking around. The second simulates the agent circumventing an obstacle to reach a certain destination, and the third tests the place cell network's ability to handle a trajectory with a loop. Each pulse from the egocentric place cells triggers the migration of the activity bump towards to the next respecting cell to track of the location. A phase reset procedure is conducted by the pulse of any egocentric place cell. If the robot travels along the same direction, multiple pulses are observed from the same egocentric place cells reporting multiple segments along that direction. The artifacts in the raw output from the egocentric place cells, as shown in figure 10, are filtered by the input port of the place cell layer thus will not cause errors in the next layer as discussed before. We use 5 as the step size for each place cell's activity level to allow clearer differentiation. A value of 10 means that a place cell is currently firing; 5 means that it was firing then decayed by a universal leak parameter of 5. We use this scheme to create a trailing tail for a better demonstration of the travel direction, shown in figure 11. We can see that the place cell network can successfully accumulate the egocentric place cell pulses to become a tracking map in an absolute spatial location given any path.

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