IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 52, NO. 1, JANUARY 2005 99 An ON–OFF Log Domain Circuit That Recreates Adaptive Filtering in the Retina Kareem A. Zaghloul and Kwabena A. Boahen Abstract—We introduce a new approach to synthesizing Class AB log-domain filters that satisfy dynamic differential-mode and common-mode constraints simultaneously. Whereas the dynamic differential-mode constraint imposes the desired filtering behavior, the dynamic common-mode constraint solves the zero-dc-gain problem, a shortcoming of previous approaches. Also, we introduce a novel push–pull circuit that serves as a current-splitter; it rectifies a differential signal into the ON and OFF paths in our log-domain filter. As an example, we synthesize a first-order low-pass filter, and, to demonstrate the rejection of dc signals, we implement an adaptive filter by placing this low-pass circuit in a variable-gain negative-feedback path. Feedback gain is controlled by signal energy, which is extracted simply by summing complementary ON and OFF signals—dc signals do not contribute to the signal energy nor are they amplified by the feedback. We implement this adaptive filter design in a silicon chip that draws biological inspiration from visual processing in the mammalian retina. It may also be useful in other applications that require dynamic time-constant adaptation. Index Terms—Adaptive filtering, artificial vision, class AB circuits, neuromorphic engineering. I. LOG-DOMAIN FILTERING D ECREASING supply voltage with integrated circuit miniaturization is increasing interest in current-mode filters. Current-mode operation offers large dynamic range if the nonlinear device transconductance is compensated for in the filter design, such that operation remains linear outside the small-signal region. The existence of such externally linear but internally nonlinear filters was demonstrated by Adams, who first designed a circuit that “when placed between a log converter and an anti-log converter will cause the system to act as a linear filter” . He named these circuits log-domain filters. The log and anti-log operations are readily realized using bipolar transistors or MOSFETs operating in weak inversion; these devices maintain logarithmic voltage-current relationships over six decades. The principle of log-domain filter design is a simple one: use current to represent the signal , voltage to represent its loga, and note that . Thererithm Manuscript received June 6, 2003; revised July 29, 2004. This work was supported by a National Institutes of Health Vision Training Grant (T32-EY07035) and by the Whitaker Foundation under Grant 37005-00-00. The work of K. A. Zaghloul was also supported by a Ben Franklin Fellowship from the University of Pennsylvania School of Medicine. This paper was recommended by Associate Editor P. Arena. K. A. Zaghloul is with the Department of Neuroscience, University of Pennsylvania, Philadelphia, PA 19104 USA. K. A. Boahen is with the Department of Bioengineering, University of Pennsylvania, Philadelphia, PA 19104 USA. Digital Object Identifier 10.1109/TCSI.2004.840097 fore, to obtain the derivative of the voltage, divide the derivative , by the signal, . That is to say, divide the of the signal, current you wish to supply to the capacitor by the current made by the transistor whose gate (or base) is connected to it. Intuitively, this division compensates for the slope of the exponential at the transistor’s operating point, such that its current changes at a constant rate. Current-division is readily realized with logarithmic elements by exploiting the translinear principle . In theory, log-domain filters have limitless dynamic range; in practice, dynamic range is limited by the bias current. Seevinck and Frey have both proposed Class AB log-domain filters that address this shortcoming; they both use two copies of the log-domain circuit to filter the differential signal , . In Seevinck’s approach, the outputs are cross coupled, each subtracting current from the others capacitor. In Frey’s approach, a current-splitter, which receives a bidirectional input current, is placed up front; it enforces a geometric mean constraint. Unfortunately, both designs suffer from distortion when the filter’s transfer function has zero gain at dc, or close to zero, due to a reduction in bandwidth and to offsets introduced by leakage currents. In this paper, we introduce a new approach to synthesizing Class AB log-domain filters. Our synthesis procedure satisfies dynamic differential-mode and common-mode constraints simultaneously. Whereas the dynamic differential-mode constraint imposes the desired filtering behavior, as in the approaches of Frey and Seevinck , , the dynamic common-mode constraint solves the zero-dc-gain problem, a shortcoming of their approaches. Specifically, we introduce a second differential equation, with its own time-constant, that imposes the desired common-mode behavior, and, in particular, we find that imposing a geometric mean constraint that is satisfied with the same time-constant that describes differential behavior results in the simplest implementation. The remainder of this paper is organized as follows. In Section II, we introduce a novel push–pull circuit that serves as a current-splitter in our log-domain filters; it rectifies a differential signal into ON and OFF paths. In Section III, taking these complementary signals as input, we synthesize a low-pass ON–OFF logdomain filter that constrains the geometric mean of its outputs dynamically. In Section IV, taking inspiration from the retina, we realize an adaptive filter by placing our ON–OFF log-domain low-pass in a variable-gain negative-feedback path. Feedback gain is controlled by signal energy, which is extracted simply by summing complementary ON and OFF signals. This application demonstrates the rejection of dc signals—they are not amplified in the feedback path nor do they contribute to the signal energy. Section V concludes the paper. 1057-7122/$20.00 © 2005 IEEE 100 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 52, NO. 1, JANUARY 2005 To determine the behavior of these ON–OFF signals subject to constraint, we observe that (1) implies this Replacing the sum of and with common-mode input signal, we have Fig. 1. Half-wave rectification. (a) Circuit implementation of half-wave rectification. Two currents I and I are compared to one another. Both currents are mirrored on to one another, eliminating most of the common mode (i.e., dc) current and driving subsequent circuitry with the differential signals, I and I . V determines the level of residual dc signal present and I . Additional copies of I and I can be made to in I drive subsequent circuitry by connecting additional transistors in parallel. (b) In a purely differential representation, the difference between I and I , I , is encoded as the difference between I and I (top). In the ON–OFF representation, one signal or the other is active, depending on the sign of the I , residual dc difference between the input signals (bottom). When I currents are inversely proportional to the common-mode input, I . Tick marks on the horizontal axis of both graphs represent units of I . = II. ON–OFF RECTIFICATION To construct our class AB log-domain filter, we first construct a circuit to divide input signals into complementary ON and OFF paths. Taking inspiration from Frey’s current-splitter , we subtract the two input currents and completely divert the difference to the ON or OFF path based on its sign. , where is the (2) and is strictly less than Thus, the geometric mean of . If , then . Conversely, if , then . Consequently, and in the first case, while and in the second case. We can see that the circuit rectifies its in. Hence, once exceeds puts around a level determined by by several , current is diverted entirely through the OFF exceeds by several , current path. Conversely, once is diverted entirely through the ON path. Whereas a conventional differential circuit would maintain current in both paths (Fig. 1(b), top), our ON–OFF design maintains current in only one path as shown in the analytical solution presented in the bottom of Fig. 1(b). of We can also determine the predicted quiescent level and when , which represents the common-mode input current level, from (2) (3) A. Implementation We implement rectification using the circuit shown in Fig. 1(a). This circuit is similar to that proposed in , but the analysis presented in that paper includes effects of , which and we ignore here. The circuit takes two input signals, , on either side, and compares them to one another. In our application (see Section IV-A), these currents represent a signal and its mean, such that current is diverted to either the ON or OFF pathways based on whether the signal lies above or below that sets the its mean. We define a current residual current level and assume a unity subthreshold slope ). Hence, the currents in the current coefficient (i.e., and . mirror can be expressed as Equating these currents to the input and output currents, we find (1) assuming subthreshold operation ( , where at room temperature and , and where signs reverse for ) and saturation . We pMOS, referenced to and as a function of can solve these equations for , , and . We note that mirroring the input currents on to one another preserves their differential signal, , . In our ON–OFF circuit, imposes which equals and a common-mode constraint on the output currents through (1). (4) when . Hence, the common-mode rejection in our circuitry is in fact not complete. Its outputs contain a residual dc component that is linearly proportional to and inversely proportional to the common-mode input signal, as shown in Fig. 1(b). Finally, because we have assumed the transistors are in saturation, our results do not apply to input currents , or (since and ). For currents above this level, the current mirrors’ output transistors enter the ohmic region, and and start leveling off. The maximum level they hence can achieve is . ON–OFF B. Simulation Results To verify our rectifying ON–OFF design, we simulated the cirand and cuit of Fig. 1(a) by sweeping the dc currents at 1. We show the relationship and recording the outputs, and , and the differenbetween the output currents, tial input, in Fig. 2, top. From the figure, we see that our simulation results replicate the theoretical prediction shown increases, in Fig. 1(b). Specifically, as the difference 1In this, and other simulations presented later, we use Tanner Tools T-Spice to simulate our circuits. We use the model file for the TSMC 0.35 fabrication process, SPICE 3f5 Level 5, HSPICE Level 49, UTMOST Level 8, released by MOSIS May 17, 2000. m ZAGHLOUL AND BOAHEN: ON–OFF LOG DOMAIN CIRCUIT THAT RECREATES ADAPTIVE FILTERING IN THE RETINA 101 A. Synthesis Procedure Our circuit design is based on the log-domain filtering approach , . To derive the circuit, first we implement complementary signaling by representing all signals differentially. Thus, (5) becomes (6) and are the ON input and output currents, and where are the OFF input and output currents. In subthreshold, these currents are an exponential function of their gate voltages (e.g., ) and so (6) becomes Fig. 2. Rectification circuit simulation. Simulation results for the rectifying and I and circuit of Fig. 1 demonstrating the relationship between I I I (top). The geometric mean of I and I is also shown as a function I (bottom). Circuit parameters: V = 2:2 V, V = 3:3 V. of I 0 0 current is diverted to the ON path, while decreasing the differcauses current to be diverted to the OFF path. In ence both cases, as soon as the difference in input exceeds even just 1 nA, current is virtually diverted to one path. Furthermore, we also see from the simulation results that when the difference between input currents is very small, the circuit maintains a small residual common-mode current in both paths. The level of this . Hence, our rectificacurrent, as shown above, depends on . tion is soft, and can be made softer by decreasing on our cirTo demonstrate the constraint imposed by cuit, we also plotted how the geometric mean of the output cur, depends on the difference in input currents, rents, . We find that is indeed small for all and rises around the region where ( Fig. 2, bottom). , is minimum Because the common-mode output, at this point, thereby maximizing the denominator in (3), our should actually fall in this reanalysis predicts that gion. This may be because we ignore in our analysis. A closer look at the circuit diagram of Fig. 1(a) reveals that as the input voltage falls on one side of the circuit, the output on that side decreases faster than the output on the other side increases. ’s effect on our circuit causes the source voltage on the one side of the circuit to have a stronger effect on output current than the gate voltage on the complementary side. Thus, if we increase away from , for example, we find that the ON current increases slower than the OFF current decreases. Hence, we . The converse is true if we find a slight decrease in move in the other diretion away from . (7) Second, we force the ON and OFF outputs, and , to satisfy a geometric mean constraint, implementing this dynamically. Thus, the product of their currents always equals , which sets quiescent output activity. This relationship is also governed by its own time constant, , and so we derive the second equation for our filter Expanding the derivative using the same subthreshold voltage–current relationship as above, we find that (8) If we express both bias currents ( (8) become and in terms of actual capacitances and , ), (7) and (9) (10) Substituting (10) into (9) to eliminate , we find that III. ON–OFF LOW-PASS FILTER We present our class AB log-domain filter synthesis procedure using a first-order low-pass filter as an example. The timedomain equations that govern the inputs and outputs of this circuit are where is the time constant. signals. and (5) If we assume that the two time constants, and , are equal, we can take advantage of the fact that . Thus, and , where and we define determine the filter’s time constant for both common-mode and differential signals. The equation then simplifies to are the input and output (11) 102 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 52, NO. 1, JANUARY 2005 Fig. 3. ON–OFF temporal filter subcircuits. (a) Subcircuit used to compute I = (I = (I + I )) I . I is subsequently used to excite the positive output node, V , and to inhibit the negative output node, V . (b) Subcircuit used to inhibit the positive output node, V , with I = (I = (I + I )) I , and to inhibit the negative output node, V , with I = (I = (I + I )) I . (c) Subcircuit used to excite the positive output node with (I = (I + I )) (I =I ) and to excite the negative output node with (I = (I + I )) I =I . Similarly (12) A CMOS circuit that is described by (11) and (12) will realize the computations needed to implement low-pass filtering in our push–pull model. By dividing the right-hand sides into two current terms that charge or discharge the filter’s capacitors (i.e., and ), we can derive the subcircuits that will realize these computations. B. Implementation Procedure Starting with the first term on the right of the equations, we construct the subcircuit shown in Fig. 3(a). Current entering this is modulated through a tilted nMOS mirror that subcircuit generates the current . For simplicity, we ignore and express all voltages in units of the thermal voltage, . Thus By setting this current, , equal to the sum of the positive and and , we can show that the curnegative output currents, rent in Fig. 3(a) is equal to the terms in (11) and (12). Specifically By setting , the current , which we use to , equals . An identical circharge up cuit on the negative side of the circuit generates a current . Taking the difference between these two currents with a current mirror (shown in Fig. 4) yields the first at the source this current terms of (11) and (12). A bias mirror keeps the drain voltages of the current mirror’s transistors similar, insuring that excitation on to one side of the circuit is matched by equal inhibition from the complementary side. The first part of the second term of (11) and (12) represents a leakage current. We implement this using a current divider that links ON and OFF sides of the circuit, as shown in Fig. 3(b). The current drawn through both sides, , is equal to Fig. 4. Log-domain low-pass filter. The complete log-domain low-pass filter circuit that implements (11) and (12). The differential signals, I and I , are inputs to the circuit, which produces the differential signals, I and I , at its output. . Hence, the current on one side of the current divider, , is This current drains charge away from the capacitor on the positive side of the circuit, and a complementary current drains charge from the capacitor on the negative side of the circuit. Hence, the first part of the second term of (11) and (12) is satisfied. Finally, the second term of (11) and (12) includes a second current that is dependent on the quiescent activity, , which determines total output activity by charging both capacitors. The subcircuit that realizes this term is shown in Fig. 3(c). Current through the nMOS transistor gated by is equal to the sum of the positive and negative output currents. Hence This node gates two nMOS transistors that dump current and ). On the positive side, back on to the capacitors ( this current is given by ZAGHLOUL AND BOAHEN: ON–OFF LOG DOMAIN CIRCUIT THAT RECREATES ADAPTIVE FILTERING IN THE RETINA 103 Fig. 5. Low-pass filter simulation. Low-pass filter simulation results with differential inputs I and I (top trace). The differential outputs are I and I (middle trace). We also show the square root of the product of these signals (middle trace) to verify the geometric mean constraint. Comparing the purely differential signals I I and I I demonstrates a 60 phase shift at 10 Hz (bottom trace). The amplitude and phase of the first Fourier component of I I , as well as the total harmonic distortion, are shown at different input frequencies (bottom). Dashed line on left curve indicates a one decade per decade slope. Circuit parameters: V = 50 mV, V = 0:4 V, V = 0:583 V, V = 1:4 V, V = 1:5 V, C = 1 pF. 0 0 If we set becomes 0 , then this current charging By defining the current as , this current satisfies the last term of (11). A complementary current charges the negative capacitor. By combining these three subcircuits, we realize all the terms in (11) and (12), yielding the complete log-domain low-pass filter circuit shown in Fig. 4. C. Simulation Results To verify our ON–OFF low-pass filter implementation, we simulated the synthesized circuit, shown in Fig. 4, which satisfies (11) and (12). We provided two 100-pA peak-to-peak sinusoidal currents 180 out of phase with one another, centered around a and . We measured mean of 110 pA, at the circuit inputs, and in simulation and took the difference between them at different input frequencies to determine how well our design would filter high frequencies and to determine the amount of distortion created by our circuit. The time-domain response of this circuit to 10-Hz inputs is and , shown in the top of Fig. 5. The differential inputs, and , that lag behind the input yield differential outputs, by roughly 60 . This relationship is best demonstrated when to the difference . comparing the difference Because, by design, we constrain the product of ON and OFF output activity with , we also show the geometric mean of the and . We see that the geometric mean is output currents, relatively flat and only dips slightly when activity switches from one side of the circuit to the other. The Fourier amplitude and phase of the circuit’s differen, at different input frequencies is shown tial output, in Fig. 5 (bottom). We see that our push–pull log-domain circuit essentially implements a first-order low-pass filter whose Hz. This corner frequency is defined corner frequency is by the filter’s time constant, which is determined by and . , and hence this In our simulation, we used values of corner frequency would correspond to an of 1.7 pA. However, when we measured in our simulation, we found it to be 0.6 pA. Because we are operating the circuit at such low currents, leakage currents could account for this discrepancy. Being such a small current, is directly affected by these leakage currents, and although we measure only 0.6 pA in our simulation, additional leakage currents in the simulation substrate may cause to appear to be 1.7 pA. Furthermore, we also find that the total harmonic distortion of the output signals reaches 6% at low frequencies and decreases with increasing frequency. Through our log-domain synthesis procedure, we have succeeded in designing a filter that remains quite linear for frequencies up to 100 Hz, which is the range of frequencies we are interested in for our biological model (see Section IV). More sophisticated 104 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 52, NO. 1, JANUARY 2005 current multiplier/divider circuits that do not require be used to achieve better performance . may IV. ADAPTIVE FILTER APPLICATION We have used the class AB log-domain filtering approach presented here to construct a circuit inspired by adaptive filtering in the mammalian retina. The retina, one of the best studied neural systems, signals the onset or offset of visual stimuli in a sustained or transient fashion . To encode these signals into spike patterns for transmission to higher processing centers, the retina has evolved intricate neuronal circuits that capture information contained within natural scenes efficiently . This visual preprocessing, realized by the retina, occurs in two stages, in the outer and inner retina, and in two complementary paths. The retina’s complementary signaling scheme is reminiscent of Seevinck and Frey’s approaches, and so we adopt the class AB log-domain filtering approach to implement a proposed model of the inner retina , . Our model for processing in the inner retina is based on the hypothesis that the inner retina adapts its low-pass and high-pass temporal filters to contrast and frequency in order to optimally encode signals . Information theory stipulates that the optimal filter for capturing information contained in natural scenes is bandpass in space and time, with the filter’s peak lying at the spatial and temporal frequencies where input signal power drops to the noise floor , . As different stimuli are presented to the retina, optimal coding requires this filter’s peak frequency to move accordingly. Thus, the retina adapts to temporal frequency to continue to convey information efficiently to higher cortical structures. Furthermore, in the case of increased contrast, which results in an increase in stimulus power, optimal filtering demands that the peak of this bandpass filter move to higher frequencies. Physiological data indeed demonstrates that the inner retina’s temporal filter realizes this adaptation to contrast—ganglion cell responses compress in time and amplitude when driven by steps of increasing contrast —by adjusting its time constant , . A. ON–OFF Signaling The second stage of visual processing begins with the bipolar cells, a class of feedforward neurons  that rectify signals received from the outer retina into complementary ON and OFF paths , , ensuring efficient information coding . These pathways are realized through a sign-reversing synapse in one path and half-wave rectification in both , . Complementary signaling is maintained in the inner retina through reciprocal inhibition between ON and OFF paths, realized by a set of narrowfield amacrine cells that ensure that only one path is active at any time. Such push–pull interactions between ON–OFF paths have been demonstrated physiologically through the existence of vertical inhibition between ON and OFF laminae . Serial inhibition  may also play a vital role in these interactions. We use our ON–OFF rectifying circuit, described in Section II, to implement the retina’s complementary signaling scheme. Using currents computed in our outer retina circuit , we define cone terminal (CT) activity as , which we compare to equal a reference current, which we define as . We set Fig. 6. Inner retina system-level diagram. Narrow-field amacrine cell (NA) signals represent a low-pass-filtered version of bipolar terminal (BT) signals and provide negative feedback on to the bipolar cell (BC). The wide-field amacrine cell (WA) network modulates the gain of NA feedback. WA receives full-wave rectified (double arrows) excitation from BT and full-wave rectified inhibition from NA. BT drives sustained ganglion cells (GCs) and the difference between BT and NA drives transient ganglion cells (GCt). such that the difference is positive to the mean value of when light is brighter ( decreases) and negative when light is dimmer ( increases). The outputs of this ON–OFF circuit represent activity at the ON and OFF bipolar terminals. Thus, this first stage of our circuit recreates computations performed by bipolar cells by diverting cone signals into complementary ON and OFF paths. We use our ON–OFF low-pass filter, described in Section III, to recreate the synaptic interactions found in the inner retina. Bipolar terminals (BTs) excite narrow-field amacrine cells (NAs) in the inner retina. Large time-constants associated with NAs make this computation analogous to a low-pass filter. Furthermore, because of the retina’s complementary signaling scheme, we implement this low-pass filter in complementary ON and OFF paths. Thus, we can simply define the inputs to our and , as ON and OFF BT activity, ON–OFF low-pass filter, derived from our ON–OFF rectifying circuit. Similarly, the and , are defined outputs of our ON–OFF low-pass filter, as ON and OFF NA activity. B. Variable Gain Feedback We propose in our model for processing in the inner retina that temporal adaptation is implemented through wide-field amacrine cell (WA) modulation of NA feedback (pre-synaptic inhibition) . Thus far, our circuit synthesis procedure presented here only computes feedforward BT to NA excitation. A system-level diagram of our complete inner retina model is shown in Fig. 6. Governed by this system diagram, we synthesize the remainder of our inner retina circuit by implementing NA to BT feedback inhibition, NA to GC (ganglion cell) feedforward inhibition, and BT to GC excitation. NA feedback inhibition is described by (13) where reflects WA activity, which is determined by the ratio of full-wave rectified BT excitation over full-wave rectified NA inhibition, as described in . To implement NA feedback inhibition on to BT, modulated by WA, we use the subcircuit shown in Fig. 7. The voltage at node represents WA activity and is the source of a transistor gated by . Thus, this activity modulates NA feedback inhibition on to BT—as voltage increases, gain, , goes down ZAGHLOUL AND BOAHEN: ON–OFF LOG DOMAIN CIRCUIT THAT RECREATES ADAPTIVE FILTERING IN THE RETINA 105 which is represented by voltage deviations below . Finally, quiescent NA activity is controlled by as discussed above. C. Simulation Results Fig. 7. Modulated negative-feedback subcircuit. Subcircuit that realizes modulation (WA) of negative feedback (NA) on to the input node (BT). WA activity, determined by voltage V , controls the source voltage of a feedback transistor, setting the feedback gain w . WA is excited by I and inhibited by wI (OFF inputs are not shown). Fig. 8. Adaptive filter. The inputs to the circuit I and I are fed to the ON–OFF rectifying circuit of Fig. 1(a), which produces three copies of its outputs (I and I ). One pair drives the ON–OFF low-pass filter. A second pair is used to excite wide-field amacrine cells (WA). The third pair is the high-pass output signal. The low-pass output signals are I and I . The log-domain low-pass filter circuitry is shown in Fig. 4. Negative feedback is modulated by node WA, which multiplies I and I by a gain, w . and as voltage decreases, gain increases. Furthermore, WA activity at this node changes with BT excitation and NA inhibition. decreases with increased current in and (not shown), thus realizing excitation of WA activity (increased gain), and inand (not shown), thus creases with increased current in realizing shunting inhibition of WA activity. Convergence of ON and OFF signals implements full-wave rectified BT excitation and full-wave rectified NA inhibition. Finally, WA nodes are coupled to one another through an nMOS diffusion network , which determines the strength of WA coupling. gated by By adding this subcircuit, we can close the feedback loop in our inner retina model, producing the final circuit shown in Fig. 8. , , and deFor the biases, the relationship between should be set equal to termine BT-to-NA gain. Ideally, for a gain of one. If , then the gain is greater than one, thus WA activity should be lower . However, if , then the dc loop gain is less than one, causing the determines residual current passed to opposite effect . acts as a reference for WA activity, the inner retina from BC. To demonstrate that WA modulation of NA feedback inhibition produces temporal adaptation, we simulated the inner retina circuit of Fig. 8. As we did not simulate an entire network, we could not exploit spatial averaging to compute mean WA activity, as the retina does. We used temporal averaging instead, which has the disadvantage of being slow, by connecting a large nF to node WA. capacitor In practice, one side of our bipolar circuit is tied to a reference voltage which sets the mean activity in the outer retina , while the other side fluctuates with light intensity. To maintain to one side this convention, we input a fixed 5-nA current of the bipolar circuit (the ON–OFF circuit shown in Fig. 1) and a that 0.125-Hz frequency modulated sinusoidal 1 nA current fluctuates around a 5-nA mean level to the other side. The carrier frequency of this signal was 55 Hz, and we used an index of modulation of 360 (defined as the ratio between the depth of modulation, 45 Hz, which represents half the frequency range, and the modulation frequency, 0.125 Hz), thus giving us a signal whose frequency cycled from 10 to 100 Hz over an 8-s period. The outputs of this bipolar circuit feed our inner retina circuit. The response of the inner retina circuit to these inputs is shown in Fig. 9(a). The input of the low-pass filter, bipolar ter, minal activity, is represented by a differential signal, in the first trace. The output of the low-pass filter is also reprein the second trace of the sented by a differential signal, figure. At the beginning and end of the cycle, this low-pass filter output is larger because of the low input frequencies, thus providing more inhibition on to wide-field amacrine cells. From the , simulation, we find that wide-field amacrine cell voltage, driven by inputs from narrow-field amacrine cells and bipolar cells, fluctuates at the 0.125-Hz modulation frequency of our input, as shown in the third trace. In regions where input frequency is low, narrow-field amacrine cell inhibition drives upwards. In regions where input frequency is high, bipolar terdownwards. minal excitation drives Because is at the source of the narrow-field amacrine cell’s feedback transistor, WA activity is below the source bias, , for our low-pass filter throughout the trace, and . By thus provides a gain, , to our feedback signal, taking the exponential of , we can directly see what this gain term is in the fourth trace. Thus, modulated NA feedback, , is larger than unmodulated NA activity, as the feedback gain, , exceeds one. This modulated feedback signal roughly matches the bipolar signal, , as shown by the overlay in the first panel. We expand the central region of this trace in Fig. 9(b). The simulation demonstrates temporal frequency adaptation since, as the frequency of the input signal changes, the system changes WA activity such that BT excitation is balanced by NA inhibition. By adjusting its time constant, our circuit design based on circuitry in the inner retina demonstrates temporal adaptation . Because we modulate the input frequency sinusoidally, we can see this adaptation for different temporal frequencies by observing the simulation results over time. This adaptation 106 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 52, NO. 1, JANUARY 2005 Fig. 9. Adaptive circuit simulation. (a) Circuit simulation of entire adaptive circuit. Circuit parameters are identical to Fig. 5, except we set V = 1:1 V to minimize the effect of dc signals on wide-field amacrine cell adaptation. V = 2:2 V. Circuit activity is represented by differential signals. Note that the voltage at WA fluctuates with the same signal frequency, 0.125 Hz, as our input frequency is modulated. Through this frequency modulation, the input frequency to the system cyles from 10 to 100 Hz over a period of 8 s. V sets the system loop gain (gain = e , bottom trace) greater than one, making the feedback (shown here as the differential signal w (I I )) larger than the unmodulated low-pass signal I I . (b) A magnified view of the first trace in (a) demonstrating adaptation in regions of higher temporal frequency. 0 matches the system’s time constant to the input, as we expect the mammalian retina to do in response to changing scenes. In addition, low-pass and high-pass signals from our circuit have a quarter-cycle phase difference and equal amplitudes over a wide range of stimulus frequencies. Thus, the circuit approximates a Hilbert transform, which has been used to model human visual motion sensing . Other visual computations, such as tracking algorithms, collision avoidance algorithms, and vision-based robotics, may benefit from this adaptation to temporal frequency, which produces a speed-invariant representation. This approach and this design may be useful in any application necessitating dynamic time constant adaptation. When changing input frequencies unbalance amplitudes in the high-pass and low-pass paths, adaptation brings these signals into balance. This adaptation remains effective until the low-pass filter’s output drops below its dc offset . Furthermore, amplification of differential signals and rejection of dc signals in our filter preserves temporal stability that was absent in earlier designs . Thus, our design presented here may be useful in other applications where adaptation and stability are 0 important. Our inner retina design corrects flaws in the design in , which failed to produce temporal adaptation. V. CONCLUSION Inspired by the mammalian retina’s complementary ON–OFF paths, we implemented log-domain filtering through a push–pull circuit that extends dynamic range without increasing power consumption. Furthermore, by modeling variable gain negative-feedback in narrow-field amacrine cells, we realized time-constant adaptation. We replicated these nonlinear temporal filtering operations in subthreshold CMOS circuits using a new log-domain synthesis procedure that extends earlier implementations of current-mode class AB circuits ,  by imposing a dynamic geometric-mean common-mode constraint. This approach simplifies the extraction of signal energy (full-wave rectification) required for adaptation and for modulation of loop gain without affecting common-mode gain or stability. Experimental test results from a retinomorphic chip that uses these circuits to recreate visual processing in the mammalian retina are presented elsewhere . 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Eng., vol. 51, no. 4, pp. 667–675, Apr. 2004. Kareem A. Zaghloul received the B.S. degree from the Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, in 1995, and the M.D. and Ph.D. degrees from the University of Pennsylvania, Philadelphia, in 2003. The Ph.D. degree was awarded in the Department of Neuroscience, where he worked on understanding information processing in the mammalian retina with Dr. K. A. Boahen. He is currently a Resident Physician in the Department of Neurosurgery, University of Pennsylvania. Dr. Zaghloul is a Member of Tau Beta Kappa and Eta Kappa Nu. Kwabena A. Boahen received the B.S. and M.S.E. degrees in electrical and computer engineering from The Johns Hopkins University, Baltimore, MD, in the concurrent Masters–Bachelors program, both in 1989, and the Ph.D. degree in computation and neural systems from California Institute of Technology, Pasadena, in 1997. He is an Associate Professor in the Bioengineering Department, University of Pennsylvania, Philadelphia, where he holds a secondary appointment in the Electrical Engineering Department. His current research interests include mixed-mode multichip VLSI models of biological sensory and perceptual systems, and their epigenetic development and asynchronous digital interfaces for interchip connectivity. Dr. Boahen received a National Science Foundation (NSF) CAREER Award in 2001 and an Office of Naval Research (ONR) YIP Award in 2002. He held a Sloan Fellowship for Theoretical Neurobiology during his Ph.D. studies. He was also awarded a Packard Fellowship in 1999. He is a Member of Tau Beta Kappa.