Speed-invariant encoding of looming object distance requires power law spike rate adaptation

Abstract

Neural representations of a moving object’s distance and approach speed are essential for determining appropriate orienting responses, such as those observed in the localization behaviors of the weakly electric fish, Apteronotus leptorhynchus. We demonstrate that a power law form of spike rate adaptation transforms an electroreceptor afferent’s response to “looming” object motion, effectively parsing information about distance and approach speed into distinct measures of the firing rate. Neurons with dynamics characterized by fixed time scales are shown to confound estimates of object distance and speed. Conversely, power law adaptation modifies an electroreceptor afferent’s response according to the time scales present in the stimulus, generating a rate code for looming object distance that is invariant to speed and acceleration. Consequently, estimates of both object distance and approach speed can be uniquely determined from an electroreceptor afferent’s firing rate, a multiplexed neural code operating over the extended time scales associated with behaviorally relevant stimuli.

Determining how nervous systems maintain perceptual invariance in the face of multifeatured sensory input is a general problem when attempting to connect sensory physiology to high-level perception. For instance, spatial parameters, such as an object’s size, distance, and orientation, are inextricably confounded in sensory images projected onto the retina (1). This becomes an even more acute problem for the encoding of dynamic stimuli (1); when presented with a temporally modulated visual stimulus, retinal ganglion cells are sensitive to as many as six different features of the input (2). In its most reduced form, a generic neuron’s conversion of input current into membrane potential is characterized by a membrane time constant (SI Text). A single response time constant endows a neuron with the ability to encode variations in stimulus intensity faithfully over one specific time scale. Because naturalistic stimuli can vary over a wide range of time scales, there will be inevitable mismatches between the rate at which the stimulus intensity changes and the temporal dynamics that define a neuron’s response. Therefore, during sensation, neurons may be highly sensitive to both stimulus intensity and the time course over which it evolves. This introduces ambiguity into the estimation of stimulus features from a neuron’s firing rate and presents a further challenge for understanding how neural systems maintain perceptual invariance.

An object moving toward an animal provides a concrete example of such a problem because two variables of interest, the object’s distance and approach speed, are both expected to influence the firing rate of a primary sensory neuron. These “looming” stimuli arise commonly in many sensory systems, including the electrosense, and appear to pose the same rate-coding dilemma for the electroreceptor afferents, whose firing rate has previously been described as sensitive to both stimulus intensity and its temporal derivative (3). By studying responses of the electroreceptor afferents to this natural problem of distance perception, we identify a mechanism for the temporal disambiguation of stimulus intensity from a neuron’s firing rate.

Looming stimuli are obviously important for survival and are processed by CNS networks of both invertebrates (4) and vertebrates (5). During environmental navigation, looming motion forms a basic component of the electrosensory signals experienced by Apteronotus leptorhynchus, particularly during prey capture (6, 7). As an object draws near the body, its presence is sensed by the fish as a perturbation to a self-generated electric field, whose amplitude drives the activity of ∼16,000 cutaneous electroreceptors (8). An approaching object, whose electric conductivity is greater than that of the surrounding water, causes the electric potential to rise locally (Fig. 1A) and evokes increases in the discharge rate of the electroreceptor afferents. Previous work has carefully detailed the change in transdermal electric potential as a function of object distance (9). Following this protocol, we implanted a transdermal recording dipole on an immobilized fish and moved a brass sphere at constant speeds along the lateral axis, toward the dipole center. Fig. 1B shows data obtained from the recording dipole with an overlay of the expected change in electric potential according to an empirically derived relationship (9). Using this model, we generated mimic looming signals corresponding to object approach for 6 cm along the lateral axis at speeds ranging from 0.5 to 5 cm/s. These signals were used as input for spike rate adaptation models and as looming stimuli while recording in vivo from electroreceptor afferents.

Fig. 1.

Object distance and the electrosense. (A) Model plot of the electric potential (measured in millivolts) produced by the self-generated electric field of A. leptorhynchus. The presence of conducting objects, such as this brass sphere (= 0.635 cm), causes a local increase in the electric potential, as measured with a transdermal recording dipole (white dots). Natural examples of relative conductors include aquatic plants, predators, and prey. (B) Recorded stimulus-induced change in transdermal potential (blue) is plotted as a function of the brass sphere’s lateral distance. The overlying red curve is the change in potential predicted by a previously existing model (9). Signals of this form are considered in the following figures.

Results

Speed-Invariant Coding of Object Distance.

We recorded from 24 electroreceptor afferents (seven fish) whose range of spontaneous firing rates (100–400 Hz) was similar to that previously reported (3). Although a function of object distance as expected, Fig. 2A demonstrates the remarkable fact that the firing rate of an electroreceptor afferent is practically independent of approach speed (regardless of object size; SI Text). In the absence of stimulation, fluctuations in electroreceptor afferent firing rate are normally distributed and thought to represent intrinsic noise (10, 11). Illustrated in the context of this variability, Fig. 2B shows the largest difference in firing rate between the looming speed responses of Fig. 2A, plotted as a function of distance. These small discrepancies in rate are clearly insignificant because they are masked by the inherent fluctuations in spiking activity. Furthermore, speed-induced variations to a distance rate code are negligible compared with the high firing rates over which the electrosensory afferents operate (3). At each point along the lateral axis, Fig. 2C shows the maximum differences from Fig. 2B plotted as a percent error of the averaged firing rate for that given distance. This measure of rate code corruption was determined for all 24 cells in response to the looming stimuli, and an average error was computed. This global error measure showed no significant trend as a function of distance (SI Text); therefore, we calculated a total average of 0.98%. At a distance of 6.25 cm from the fish’s skin, the brass sphere from Fig. 1 causes a change in transdermal potential of 0.15 μV, a signal weaker than the limit of behavioral detection (12). The percent error measured at this point is not significantly different from when the sphere is 0.25 cm away, causing a change in transdermal potential of 350 μV. This confirms that minute discrepancies in firing rate, introduced by approach speed, are no more deleterious for a distance rate code than endogenous noise acting in the absence of stimulation.

Fig. 2.

Electroreceptor afferents encode object distance, invariant to approach speed. (A) Averaged responses of a single electroreceptor afferent to looming stimuli with approach speeds spanning an order of magnitude (0.5–5 cm/s). Remarkably, the firing rate is a function of object distance only and is invariant to approach speed. (B) Maximum difference in firing rate for the approach speeds of A is plotted as a function of distance. The distribution of fluctuations in spontaneous firing rate, centered about the mean, is shaded, and the value of 1 SD is marked by the green line. This value (21.33 ± 5.5 Hz) was previously determined as an average over many electroreceptor units (3) for a bin width of 32 ms. Note that our exemplary comparison with these data is justified because our firing rates were computed using a smoothing kernel of comparable duration (Materials and Methods). (C) At each point along the lateral axis, the differences from B are weighted as a percent error of the average firing rate at that distance. The error is small and essentially constant, highlighting the insignificance of any speed-related differences in firing rate.

Speed Invariance Arises from Power Law Adaptation.

The electroreceptor afferents are not passive rate coders and show strong spike rate adaptation in response to sustained stimulation (13, 14). Spike rate, or spike frequency adaptation (SFA), is a ubiquitous and conserved feature of neural processing, which has been well documented in systems ranging from invertebrate sensory neurons to mammalian cortex (15, 16). A generic model of SFA consists of cellular- or circuit-level mechanisms that integrate the response of a neuron and exert negative feedback to control its output firing rate (17, 18). The different types of SFA observed experimentally are dictated by the specific form of the integrator, which is often described as a single exponential process (15, 19). As an example, exponential SFA permits the electroreceptor afferents of A. leptorhynchus to act as selective filters for communication signals (14). However, many neurons display multiple adaptive time scales, indicative of multiexponential or power law adaptation (18, 20–25). A wide range of adaptive time scales have also been identified in the electroreceptor afferents (13), prompting a demonstration that power law dynamics can account for the experimentally observed SFA (18). Therefore, we investigated two existing SFA models of the electroreceptor afferents in an attempt to understand the observed looming speed invariance.

Fig. 3A shows the looming responses of an existing model of electroreceptor afferent activity that incorporates an exponential form of adaptation (10). Previous experiments and modeling studies have shown that the time constant of exponential adaptation is very short (≈10 ms) and can successfully account for spontaneous interspike-interval correlations, as well as responses to constant intensity stimuli and high-frequency communication signals (10, 14, 26). However, when processing looming signals, such rapid adaptation is not well suited for accurate distance estimation (SI Text). In fact, regardless of the choice of time constant, the exponential adaptation model’s firing rate is a function of both object distance and approach speed, confounding estimates of these values (SI Text). In this case, a decoder of firing rate must extract information about both the spatial and temporal features of a moving object from a scalar value, a problem for which no invertible mapping exists. Because the firing rate is a nonseparable function of speed and distance (SI Text), it is not immediately clear how this response could be decoded so as to enable the accurate electrolocation observed in behavioral experiments (6, 7). Therefore, to account for the weak SFA observed over longer time scales (13), we tried a model of SFA that incorporates a power law (18). Fig. 3B shows that the model responses, like the data, only encode distance information and are nearly invariant to approach speed. Power law adaptation is able to transform a neuron’s response to nonstationary looming signals and permits the formation of a well-defined map between changes in firing rate and the distance of a looming object. As shown in Fig. 3C, the firing rate of the electroreceptor afferents is also a linear function of looming stimulus intensity until the signal becomes very strong and the response saturates. The power law adaptation model does not incorporate a saturating component and maintains a linear encoding for all intensity values (Fig. 3C). This adaptation model is based on a power law function with an exponent of −1 (18). Below, we demonstrate analytically that this is the only form of adaptation able to modify a neuron’s response to looming stimuli such that the firing rate is independent of speed. Using the same formalism, we show why the outputs of exponential adaptation models depend on speed, explaining the divergent responses seen in Fig. 3A (SI Text). The exponential adaptation model is not capable of rescaling the firing rate in response to a looming stimulus and cannot linearly encode stimulus intensity (Fig. 3D).

Fig. 3.

Power law adaptation generates speed-invariant responses. (A) Responses of an electroreceptor model with exponential adaptation (10) to our looming stimuli. Significant speed-induced discrepancies in the firing rate occur for distances less than 2 cm, a range important to the fish during hunting and tracking behaviors (6, 7). Because the estimated time constant derived from exponential models of adaptation (≈10 ms) performed poorly, the model responses illustrated here were obtained using the best possible time constant of adaptation for our range of speeds (420 ms; Inset) which was chosen to yield the smallest possible differences in firing rate (SI Text). (B) Firing rates of an established power law adaptation model (18) to the same stimuli. In this case, the firing rate is essentially a function of looming object distance only. (Inset) Model response (blue) overlaid onto the data of Fig. 2A (gray) with superb agreement. (C) Electroreceptor responses (gray) and the power law model response (blue) are plotted against the intensity of the looming stimulus, displaying a linear relationship. The dashed purple line indicates where the data began to saturate and thus deviate from linear coding. (D) Exponential adaptation model responses are plotted as a function of stimulus intensity, clearly demonstrating the failure of exponential adaptation to produce linear coding.

Many studies describe the role of SFA as a means to filter out static and low-frequency components of a stimulus (14) or to regulate a neuron’s response to prolonged stimulation at various intensities (24, 25, 27). With a few notable exceptions (22, 28, 29), little has been done to investigate the role of SFA in processing natural, nonstationary signals. Our results show that power law adaptation implements a real-time sensory transformation that shapes a neuron’s transfer function to achieve a systems level coding goal. By removing the effects of speed, this transformation permits a primary afferent’s firing rate to serve as a foundation for dynamic distance perception. Furthermore, although its effects on the response are neutralized, the approach speed can still be uniquely determined from the firing rate as described below.

Estimating Approach Speed from Firing Rate.

The derivative of the electroreceptor afferent firing rate (λ), with respect to time, is given by , where is the speed of the looming stimulus. Because λ is independent of speed, so is its spatial derivative. Therefore, at a fixed distance, the temporal derivative of the firing rate is directly proportional to speed. For illustrative purposes, three different looming responses are plotted as a function of time in Fig. 4A. For each curve, a linear approximation to the tangent was evaluated at a distance of 1.5 cm: Faster approach speeds produce steeper slopes. When evaluated at one particular distance, the firing rate’s temporal derivative () provides a simple, linear encoding of the object’s speed (Fig. 4A, Inset). Fig. 4B shows as a function of lateral distance for all the looming speeds considered. Because is a function of both distance and speed, it can achieve the same value for different combinations of these two variables. Therefore, unambiguously extracting looming speed from relies on a downstream computation: A decoder must use the distance information contained in the firing rate to calibrate the speed estimate. Provided this occurs, an invertible mapping from to the object’s approach speed can be constructed. Physiological mechanisms exist that are well suited to decode temporal derivatives, such as synapses endowed with short-term depression and a variety of feed-forward and recurrent neural networks (30).

Fig. 4.

Estimating approach speed from the firing rate. (A) Electroreceptor responses from Fig. 2A for 2-, 3-, and 5-cm/s looming stimuli are plotted as a function of time (truncated for >220 Hz to accommodate the Inset). Linear approximations to the tangents of these curves were evaluated at a distance of 1.5 cm, whose corresponding firing rate is marked by the purple dashed line. Clearly a faster moving object causes a proportionally larger temporal derivative of firing rate. (Inset) Plot of the tangent slopes, evaluated at 1.5 cm, for the six speeds considered. At each fixed distance, there exists a linear relationship between the firing rate and speed. (B) Temporal derivative of all responses from Fig. 2A, plotted as a function of distance. The temporal derivative of the firing rate is strongly influenced by the speed of motion as well as object distance. As illustrated by the black dashed line, a given value of the firing rate's temporal derivative will intersect the different curves generated by multiple approach speeds. Thus distance information must also be considered in order to avoid ambiguity when decoding looming speed.

Analysis of Speed Invariance.

To investigate the role of adaptation in speed invariance further, we sought to understand the formal importance of the power law in the model proposed by Drew and Abbott. The neuron’s membrane potential is described by a leaky-integrate-and-fire neuron as follows:

where is the membrane time constant; is the bias current; is the stimulus-induced current; is the firing rate of the neuron; and is the adaptation kernel, which is convolved with the firing rate response. This neuron spikes when the membrane potential reaches a fixed threshold and then resets to zero. Consistent with Eq. 1, it has been shown that the electroreceptor afferents express SFA through the action of a subtractive, hyperpolarizing current (14). In any case, the following result pertains to the adaptation current, , whose dependence on speed does not rely on whether adaptation is subtractive or divisive.

We show that there exists a unique adaptation kernel that enables the output of a generic neuron to encode looming distance (), regardless of the approach speed (). From our experimental data and model simulations, it is clear that power law adaptation removes the dependency of on approach speed. We have seen that the firing rate is a linear function of stimulus intensity (i.e., object distance), so let . Above, looming distance was implicitly defined as . Through a simple change of variable, we reduce the problem to studying the net current as a function of the distance traveled (). As such, the temporal dynamics of Eq. 1 are determined by

This illustrates that there is only one suitable form of adaptation kernel that can be used to achieve the observed speed invariance: , which yields

The reader can verify that the choice of results in a subtractive current that is simply a Hilbert transform of the stimulus, a common technique in signal processing to extract the envelope of a signal (31, 32). Interestingly, it has been shown that multiple adaptive time scales contribute to the encoding of low-frequency envelopes generated by whisker motion in the rat vibrissae pathway (22). It is also worth noting that power law adaptation has been observed with exponents other than −1 (20). Because the temporal disambiguation of stimulus intensity requires a unique adaptation kernel, it seems likely that different forms of power law adaptation correspond to different system-specific processing goals.

Realistically, looming motion may not always occur at a constant speed, an assumption the above argument relies on. The simple dependence of the net current on the stimulus suggests that invariance may arise more generally (i.e., for nonconstant speeds). Therefore, we tested looming stimuli that accelerated or decelerated during approach. Under these conditions, we demonstrate that neurons displaying power law adaptation can still maintain invariant representations of object distance (data, simulations, and theory are provided in SI Text).

Biophysics of Adaptation.

The success of adaptation in the above sensory transformation depends on its ability to act over different stimulus time scales. In general, adaptive processes can operate over many time scales, from tens of milliseconds to minutes (13, 33). This capacity may arise from the action of numerous biophysical mechanisms, such as voltage- and ion-gated channels (15, 17, 19, 34, 35), ion depletion and the action of electrogenic pumps (28, 36), and synaptic transmission and short-term synaptic plasticity (28, 37), as well as the inactivation kinetics of ion channels (38–41). Power law adaptation in spider mechanoreceptor afferents has been simulated with Hodgkin–Huxley (HH) neurons, suggesting that the voltage-dependent gating of sodium and potassium channels forms the biophysical basis of the power law transformation (20). We confirmed that an HH neuron with standard parameters is capable of producing the observed speed invariance for the looming signals tested in our experiments (SI Text). All these processes are common features of neural transmission, suggesting that power law dynamics may exist as an emergent property of interacting biophysical sources, encompassing a wide range of characteristic time constants. On the other hand, the fast exponential adaptation present in the electroreceptor afferents is likely controlled by a single source: the action of a voltage- or calcium-sensitive potassium channel (14). Although technical issues prevent us from conclusively demonstrating the subcellular mechanisms responsible for the different forms of adaptation, one thing is clear: The exponential and power law forms function cooperatively. While the fish navigates its environment, the amplitude modulations caused by electrolocation targets will not recruit fast exponential adaptation, whose cutoff frequency of 23 Hz exceeds their low-frequency content (14) (SI Text). However, when a conspecific emits a transient communication signal, fast exponential adaptation ensures its selective encoding over electrolocation targets, setting a momentary precedence for social information. In this manner, the two forms of adaptation have effectively partitioned a stimulus frequency space, each operating in its own regime.

Discussion

Power laws appear at many levels of biological description, from single-channel kinetics to human psychophysics (18). The scale-invariance property of the power law likely explains its ubiquity, because it can easily conform to a large range of stimulus time scales present in an animal’s environment (24, 25, 42). By permitting the time scales of the stimulus to determine the dynamics of the response, power law adaptation can actively rescale a neuron’s transfer function, imparting flexibility to a rate-coding system. In the case of the electroreceptor afferents, adaptive rescaling allows a rate code to transmit distance information about a looming stimulus reliably, regardless of its approach speed. Where exponential adaptation fails, a power law transformation provides an elegant solution to an otherwise conflated sensory estimation problem. This is similar to the notion of a multiplexed neural code (43), in that it enables disambiguation of stimulus features that cannot be discriminated on a single response time scale. However, instead of different stimulus features being encoded into different time scales of the spiking response, distance and speed are simultaneously encoded through the action of adaptive processes that operate over extended time scales.

When an animal selects a behaviorally appropriate response to an approaching object, accurate estimates of speed and distance are fundamentally important. This has been made particularly clear in the looming responses of the locust, where representations of distance and approach speed form the substrates of a time-to-collision computation (4). Different target cells in the electrosensory lateral line lobe (ELL) appear well suited to begin extraction of the distance and speed information present in the output of the electroreceptor afferents. ELL pyramidal cells can be divided into two broad classes: those that respond in a near-linear manner to the electroreceptor afferent firing rate (44) and highly nonlinear cells that are very sensitive to changes in firing rate (44, 45). Therefore, we propose the linear class of pyramidal cells relays information about an object’s distance from the skin, whereas the nonlinear class initiates extraction of approach speed from the temporal derivative of the firing rate. The midbrain target of the ELL pyramidal cells is the torus semicircularis, a structural analog of the mammalian inferior colliculus. In turn, the torus provides massive input to the optic tectum (46), an analog of the mammalian superior colliculus. “Collision-sensitive” neurons have been discovered in the optic tectum of frogs (47), suggesting that the differential responses of ELL pyramidal cells initiate distinct streams of information, which serve as the basis for tectal neurons to compute and initiate appropriate motor commands in response to looming stimuli.

It was recently demonstrated that a class of retinal ganglion cells can register the location of longitudinally moving objects independent of their speed (48), achieving the same effect that we have described for the electroreceptor afferents. It would be interesting to determine whether the biophysical mechanisms described in this paper also exploit the scale-free nature of power law dynamics or whether another sort of computation can result in speed-invariant localization.

Neural implementation of efficient coding strategies has been the subject of intense investigation. Unfortunately, it is not always easy to recognize or confirm that a particular coding strategy has been achieved. In the work presented here, understanding the advantage for neural coding was facilitated by a well-understood context: precise knowledge about the nature of the inputs (9), the corresponding behavioral goals of the animal (6, 7), and the suggested encoding scheme (3). The role of power law adaptation should be further explored in response to realistic, nonstationary inputs, including the envelope signals generated in the visual (49), auditory (50), and somatosensory systems (22). The range of frequency content in these envelopes will draw out different time scales of adaptation at the cellular and circuit levels that may support power law transformations. Can scale-free dynamics also optimize coding strategies in these sensory systems? How about higher levels of the CNS? Answers to such questions should enhance our understanding of the different functional purposes that adaptation and power law relationships serve for efficient neural coding.

Materials and Methods

A standard surgical protocol was performed to expose the hindbrain of A. leptorhynchus, and all procedures were approved by the Animal Care Committee at the University of Ottawa (full experimental details are provided in SI Text). The fish were immobilized and mounted into a large tank of 27 °C water with the electric conductivity kept between 120 and 150 μS/cm. Glass micropipettes (filled with 3 M potassium acetate, resistance of 90–120 MΩ) were advanced through the cerebellum to take extracellular recordings from electroreceptor nerve afferents in the deepest layer of the ELL (8). Firing rates for electrophysiological data, as well as for the models, were computed by convolving binary spike trains with a causal 50-ms exponential kernel. Averaged firing rates were then computed from repeated presentations of the looming signal (10–20). All analysis was performed using custom scripts and available functions in MATLAB (MathWorks). We recorded the change in transdermal potential resulting from a looming brass sphere moving toward the recording dipole center. The brass sphere was attached to a thin plastic rod and mounted to the mobile platform of a Parker LP28 linear actuator, controlled by a Parker ViX 250 IM microstepping drive. The looming sphere saturates the tiny receptive fields of the electroreceptors directly in front of it (8). During in vivo recordings, finding the exact center of the tiny electroreceptor receptive field can be challenging. Furthermore, recordings from the fine electroreceptor afferents are difficult to maintain while trying to align the motor manually and position it within a given electroreceptor’s receptive field. Therefore, using the stimulus shown in Fig. 1B, we directly modulated the amplitude of a fish’s electric field to simulate electrosensory looming stimuli. This guaranteed that the looming approach was simulated exactly over the receptive field center of the particular electroreceptor afferent whose activity was being recorded.