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1 Department of Neurobiology, The Weizmann Institute
of Science, Rehovot 76100, Israel,
2 Department of Physics and
3 Graduate Program in Neurosciences, University of California at
San Diego, La Jolla, CA 92093 and
4 Institute for Theoretical
Physics, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
Address correspondence to Ehud Ahissar, Department of Neurobiology, The Weizmann Institute of Science, Rehovot 76100, Israel, email: ehud.ahissar@weizmann.ac.il, or to David Kleinfeld, email: dk@physics.ucsd.edu.
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Abstract |
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Introduction |
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). However, the
brain also contains an enormous amount of recurrent and feedback
connections at various levels. The possible function of recurrent
connections, i.e. connections between neurons at the same level of
processing, has been intensively investigated. One of the more
attractive suggestions for such a function has come from analogies
with the statistical mechanics of many particle systems:
recurrent networks could function as associative networks for memory
(Hopfield, 1982; Amit, 1989), as well as
continuous attractors for sensory coding (Ben-Yishai et al.,
1995) or motor
control (Seung, 1996).
These networks, in their most basic form, are largely homogeneous
so that all neurons participating in such a network have a similar
role in computation.
Feedforward and recurrent computation schemes can be reconciled by
viewing information as being passed from one processing station to
another in a feedforward manner and processed at each station by
recurrent networks (Fig. 1a). However, the
description of brain architecture is not complete without inclusion
of its third major component — large-scale feedback connections.
Feedback connections, which feed the output of the receiving
areas back to the transmitting areas, occur at all levels (Fig.
1b). Cortico-thalamic
feedback connections are perhaps the most intensively studied example
of this kind. Feedback connections, however, occur not only between
cortex and thalamic nuclei, but also between cortex and brainstem,
between cortical areas that are connected via feedforward
connections, and from motor output nuclei back to the cortex; see
Kleinfeld et al. (Kleinfeld et al., 1999) for a review
on the vibrissa sensorimotor system.
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). Servo loops
are systems whose aim is to keep an output variable within a
predetermined range. These loops use feedback to determine the actual
state of the output variable and a comparator circuit to compare
this value with the target value. For example, keeping a stick
upright using one finger requires a servo operation. A servo loop
is a particular type of closed-loop system. In general,
closed-loop systems can be used to accomplish various decoding,
transformation and control tasks. Such tasks are routinely performed
in the brain.
The architecture of closed-loop systems lies between that of feedforward and recurrent networks. Like in feedforward networks, but unlike in recurrent networks, the flow of information in closed-loop systems is well delineated. Like in recurrent networks, but unlike in feedforward networks, information flows in both directions, i.e. from input to output and back. Thus, closed-loop circuits provide a substrate for computations that cannot be done with purely feedforward or recurrent configurations. One example is iterative transformations from one set of neuronal variables to another, as may occur in the encoding and processing of sensory inputs.
Closed-loop dynamics can be found at all levels of neuronal function. At the molecular level, the activity of a biochemical process can be suppressed (negative feedback) or enhanced (positive feedback) by the end product of that process. Similarly, at the cellular level, the opening of ion channels is a function of the membrane potential, which in turn is affected by ion channel opening. At the circuit level, the activity of individual neurons influences neighboring cells, whose activity in turn modulates that of the original cell. At the system level, e.g. the level of cortical areas and sub-cortical nuclei, each neuronal circuit affects several other circuits whose output ultimately feeds back on the original circuit. At the behavioral level, sensory input guides the motor response, which in turn updates the input to the sensory system. We focus here on the circuit, system and behavioral levels.
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Computations Performed by Neuronal Closed Loops |
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Closed loops provide an elegant solution to control problems that involve different types of variables, such as the control of mechanical variables by neuronal variables. To illustrate closedloop control, we will consider two schemes that are implemented by a ‘low-level’ loop, the stretch-reflex loop, under ‘high-level’ descending control.
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Examples of Motor Control by Closed Loops |
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). The pairs of muscles
are arranged as synergists, i.e. muscles that close the joint, and
antagonists, i.e. muscles that open the joint. There are two forms of
feedback sensors at a joint. The first are spindles, which encode the
degree of muscle-stretch. To the extent that the angle is
proportional to the stretch, the spindles are reporters of the angle
of the joint. The second are Golgi tendons, which report the force
produced by the muscle or, equivalently, the torque. Both of these
receptors form a largescale feedback path from the joint back to the
motor neurons (Fig. 2b, c). For simplicity, we
will consider only positional feedback.
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0 in Fig.
2b
), generated by higher
neuronal circuits. The absolute position of the joint ( in Fig. 2b), as encoded by
the spindles, is compared with the command (
operation in Fig.
2b) and used to generate a
control signal of the form d/dt = G(–0). This signal, in turn,
drives the synergistic motor neurons and its negated form (mediated
by inhibitory interneurons) drives antagonist motor neurons. Under
steady-state conditions, the two muscle groups integrate the control
signal. For a sufficiently large gain (G in Fig. 2b), the angle of the joint
will converge on the desired angle, i.e. 
0.
A second feedback scheme, which is of relevance to our discussion
on the vibrissa sensorimotor system, considers the periodic
modulation of the angle of a joint. This control scheme makes
use of a descending signal that oscillates in time
(cos2pf0t in Fig. 2c). The position of the
joint (cos[2
f0t + 
] in Fig. 2c) is mixed with the control
signal, as could occur by neurons or small networks of neurons that
use their threshold properties to multiply their inputs [X in Fig.
2b; see Ahissar
(Ahissar, 1998) and Ahrens
et al. (Ahrens et al., 2002)]. The
spectrum of the mixed signal contains the difference between the
desired frequency and the actual frequency (f –
f0), as well as the sum of these frequencies. The
low (difference) frequencies are extracted from the mixed signal by
the low-pass filtering properties of the involved neurons. The final
signal contains a constant (Gsin in Fig. 2c), as well as a term, which
for small frequency differences (i.e. f f0) is proportional to
the phase slippage [O{(f –
f0)t} in Fig. 2c]. This final signal
is used to drive a local oscillator in the spinal cord (~ in
Fig. 2c), for which the
oscillation frequency is a monotonic function of the input. For
open-loop gains (i.e. accumulated gain along the loop) around 1, the
frequency of the local oscillator will be driven to match that of the
control signal, so that under steady-state conditions f =
f0 and the final signal is a constant with no phase
slippage. Thus there is a constant phase difference, , between the reference and spinal
oscillators, which is a monotonic function of (f –
f0)/G. By combining feedback schemes for
both position and oscillation control, a joint can oscillate around a
desired position, as is required for certain motor tasks, such as
walking.
Transformations from one variable to another are required not only for motor control, but also for sensory processing. Perceived entities are composed of a variety of physical variables of different types and dimensions. These physical variables should be transformed to neuronal variables. Hence, sensory acquisition and processing must employ a variety of transformations. The first transformations occur already at the level of the sensory receptors during transduction. These transformations are implemented by a variety of closed loops, mostly at the molecular and cellular levels. Later transformations usually transform featurebased codes to more abstract codes that are used for integration and motor control.
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Closed-loop Computations in the Vibrissa Sensorimotor System |
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) and persist for bouts
up to 4 s. Interestingly, animals will whisk at essentially a
constant frequency over the entire period of the bout, as seen in the
rectified mystatial electromyogram (EMG) (Fig. 3b), a measure of vibrissa
position. They can then whisk again at the same or a different
frequency, but again hold the frequency constant (Fig. 3c) (Berg and Kleinfeld,
2002).
Thus, whisking appears to be driven by an accurate pacemaker,
with a frequency that varies between whisking bouts, but that
is constant within a bout, or alternatively, different pacemakers
control whisking at different bouts. The distribution of the
whisking frequencies generally lies between 5 and 15 Hz (Fig.
3d) and the spectral purity
of the whisking is independent of frequency. The accuracy of whisking
within each bout is suggestive of phase-sensitive detection as a
decoding scheme of vibrissae position (Kleinfeld et al.,
1999). This decoding
can, in principle, be accomplished by open or closed loops (Ahissar,
1995, 1998;
Ahissar et al., 1997).
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).
This system also contains feedback loops that enclose various
processing stations (i.e. sensory thalamus and cortex) within
closed anatomical loops. Is local computation between these
stations dominated by feedforward or closed-loop dynamics? With
feedforward processing, sensory signals should propagate to the
cortex via brainstem and thalamic ‘relay’ neurons. Thus,
vibrissa-locked activities of somatosensory cortical neurons should
lag whisking activity. However, during epochs of vibrissalocked
oscillatory activity in freely behaving rats (Fee et al., 1997), as
well as in awake restrained rats (Brecht and Sakmann, 2001),
cortical neurons show a full range of phase relations relative
to the angle of the vibrissae. Further, in awake rats, the phase
of thalamic oscillatory activity lags the phase of both brainstem
and cortical activity (Nicolelis et al., 1995). While
inconsistent with a feedforward scheme, these observations are
consistent with a closed-loop computation scheme, in which cortical
oscillatory activity is not driven by sensory events but is generated
independently in the cortex and is used to decode, or to
‘measure’, the input periodicity.
The existence of a central mechanism that ‘measures’ the input
periodicity was proposed by Mountcastle and colleagues in the 1960s
to explain their observations from the primate somatosensory system
(Talbot et al., 1968). The
possibility that such a mechanism exists in rats was investigated by
testing the predictions of several potential mechanisms in
anesthetized rats (Ahissar et al., 1997, 2000, 2001a; Sosnik et
al., 2001) and
is reviewed elsewhere (Ahissar and Arieli, 2001; Ahissar
and Zacksenhouse, 2001). The results
of these experiments suggest that one of the two major
thalamocortical systems, the paralemniscal system, contains many
parallel loops that function as phase-locked loops (PLLs). A PLL is
an algorithm for temporal processing with periodical signals,
discovered by electrical engineers in the 1930s (Bellescize, 1932) and is
considered to be an optimal temporal decoder. It can be implemented
by software, electronic circuits (Gardner, 1979), single
neurons (Hoppensteadt, 1986), or
neuronal circuits (Ahissar and Vaadia, 1990; Ahissar,
1998).
The elegance of the PLL emerges mainly from its adaptive operation,
which is a direct outcome of its closed-loop design (Gardner,
1979; Ahissar,
1998; Kleinfeld
et al., 1999). One
implementation of a PLL was presented above to describe motor control
of the skeletal joint (Fig. 2c). Other neuronal
implementations of PLLs could, in principle, occur all over the
nervous system. In particular, a sensory PLL for decoding vibrissal
temporally encoded information could be implemented across
thalamocortical loops, by using cortical oscillators, cortical
inhibitory neurons and thalamic ‘relay’ neurons, where the last
are hypothesized to function as phase detectors (Fig. 4) (Ahissar
et al., 1997; Ahissar and
Arieli, 2001; Ahissar and
Zacksenhouse, 2001). For such
implementations, which consist entirely of spiking neurons,
discrete-time representations are probably more appropriate than
continuous-time representations (Ahissar, 1998). This is
particularly true for the vibrissal system, in which computations
involve neurons that fire one or few spikes per whisking cycle.
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Decoding by PLLs: Theory, Predictions and Tests |
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) as well as
motor cortex (Kleinfeld et al., 2002) are locked to
motion of the vibrissa over the full range of phases, 0 to 2, in the awake, whisking
animal. Second, many independent cortical oscillators exist in the
somatosensory cortex of anesthetized rodents, each exhibiting a
different spontaneous frequency and each oscillating independently
from the others when no sensory stimulus is applied (Ahissar et
al., 1997) [see also
(Ahissar and Vaadia, 1990; Schoner et
al., 1992)].
This parallel-processing scheme entails specific predictions —
predictions that emerge from the PLL equations.
The equations of a discrete form of a linear PLL (Fig. 4), in the
absence of noise, are:
 |
(1) |
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(2) |
D = 0, a is a constant,
and D(n) is the temporal delay between the two inputs,
tosc and tbs (Fig. 4) to the phase detector,
and is given by:
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(3) |
Ti(n) is the period (1/frequency) of the input, and
To(n) is the period of the oscillator. The
latter period is given by
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(4) |
is a constant.
At steady-state, D(n + 1) = D(n) and thus the PLL is
locked [To = Ti, in equation
(2)] and, from
equations (1) and (4),
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(5) |
is the ‘open-loop
gain’ of the circuit.
Thus, with linear PLLs at steady-state, D is linear with respect
to the term (Tc – Ti). The latency of
the cortical oscillator (tosc –
tstim, where tstim is stimulus onset time)
equals, from equation (3):
 |
(6) |
 |
(7) |
D + constant. This relationship, together with equation
(5), provides several testable
predictions.
The first prediction is that steady-state response latencies, in
thalamus and cortex, should increase with increasing input
frequencies [fi = 1/Ti; for any given
Tc, D will increase with decreasing
Ti, see equation (5)]. This was indeed observed
in the thalamic and cortical stations of the paralemniscal
system, i.e. POm and layer 5a, respectively (Ahissar et al.,
2000).
The steady-state response phases of POm and layer 5a neurons
increase with the stimulus frequency (Fig. 5a). The response
phases increase not only due to the decreased stimulus period,
but also due to an explicit increase in response latencies, as
demonstrated by the distributions of onset latencies among these
neurons (Fig. 5b).
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is that, when
the input frequency is constant, the spread of thalamocortical
latencies should be determined by the spread of the intrinsic
periods (Tc) of cortical oscillators. The periods
Tc cannot be measured in vivo, since the
oscillators always receive inputs as a consequence of spontaneous
activity. Nevertheless, to the extent that the spontaneous drive is
similar across different recordings, the spread of spontaneous
oscillations may provide an approximation to the spread of
Tc. Spontaneously oscillating neurons, which could
potentially function as oscillators in vibrissa-related PLLs, have
been previously recorded in anesthetized rats and guinea pigs
(Ahissar et al., 1997). To test the
above prediction, we re-plotted the frequency distribution of
the rat oscillating neurons as a period distribution, in the
range that is relevant for decoding whisking frequencies (Fig.
5c, black broken
curve). With the PLL scheme described above, Tc
should be smaller than the maximal decodable Ti. Thus,
to cover the range of spontaneous oscillatory periods which could
be relevant for decoding whisking frequencies, we depicted
periods 20 < Tspont < 120 ms, corresponding
to frequencies between ~8 and 50 Hz. Within this range, the
distribution of spontaneous oscillating periods contains a mode whose
spread approximates that of response latencies in POm and Layer 5a
(Fig. 5c), as predicted
by the PLL model. This mode contains spontaneous oscillatory periods
between 50 and 105 ms, corresponding to frequencies between 9.5 and
20 Hz. The striking overlap between the spread of
Tc and that of tD (Fig. 5c) could be interpreted to
imply that the hypothetical thalamocortical PLLs function near
their linear regime — equation (5).
According to the above results, temporal decoding in the paralemniscal
system is accomplished by sets of many parallel thalamocortical
PLLs, each with a different working range, i.e. a range of input
frequencies that can be decoded by that PLL. If this is the
case, then during natural whisking the spread of cortical phases
should exhibit the accumulation of the spreads of cortical
Tcs and of whisking frequencies. This is consistent
with cortical data from freely moving rats (Fee et al., 1997). Cortical
neurons phase-lock to whisking movements with different phases
(Fig. 6). While the ensemble
vector has a phase of about /4 relative to the retracted phase of the mystacial EMG
activity, the entire population of single units cover the entire
range of possible phases (Fig. 6b). The cortical phase
distribution observed during free whisking (Fig. 6b) resembles that observed
in anesthetized rats during vibrissa stimulation at whiskingrange
frequencies, i.e. frequencies between 5 and 11 Hz (Fig. 5a, right panel,
gray dots). In both cases, response phases distribute between 0
and 2 and response magnitudes
are larger at low response phases, consistent with the PLL
model.
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), perhaps based on
their specific loop affiliation. As described above, we
speculate that, for every neuron, its response phase lag depends on
the intrinsic period (Tc) of the oscillator driving
the loop containing that neuron. If this is the case, these phase
lags contain information about the whisking frequency. The same
information can be extracted from the spike-counts of cortical
neurons. Both signals are likely to be noisy at the cortical level
and the system may combine both sources of information to improve the
fidelity of the representation of the whisking frequency. To
compute the phase lag, the exact reference of whisking onset time
has to be available. This suggests a possible role for the
lemniscal system, in addition to its other functions, within this
temporal decoding scheme: the lemniscal system, which responds at a
fixed latency at all frequencies within the whisking range
(Hartings and Simons, 1998; Ahissar et
al., 2001a; Sosnik et
al., 2001),
can provide the required reference signal. Comparison of the
outputs of the lemniscal and paralemniscal systems, say at layer
2/3 of the cortex (Ahissar et al., 2001a), can provide a
measure of the desired phase-lag. Further, the reference signal
conveyed by the lemniscal system might be useful in computing the
contact angle when the vibrissae touch an object during
protraction. Thus, an outline of the first stages of the temporal
decoding scheme can look like that depicted schematically, and in
abstract form, in Figure 7. Note that (i) the
paralemniscal loop in fact contains many loops in parallel and (ii)
only lemniscal signals that relate to this computation are depicted,
ignoring other, parallel, lemniscal processes.
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Further Predictions for Thalamocortical PLLs |
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The last prediction depends on the actual set point of the
thalamocortical PLLs and thus is not a critical prediction (Ahissar,
1998; Kleinfeld
et al., 1999). Yet, the data
collected so far indicate thalamocortical set points for which such a
dependency is expected (Ahissar et al., 1997; Ahissar and
Arieli, 2001).
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Possible Alternative Mechanisms |
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; Buonomano and
Merzenich, 1995). The data we
collected, however, are inconsistent with such feedforward mechanisms
(Ahissar and Arieli, 2001).
Another possibility is that vibrissa position is not encoded by
temporal cues. For example, different ganglion or brainstem neurons
might be associated with different phases along the whisking path,
such that the identity of the activated neuron indicates the position
(angle) of the vibrissa (a ‘labeled-line’ coding scheme). Or,
alternatively, population of neurons might encode vibrissa position
in their ensemble firing rate. Unfortunately, a systematic
investigation of the encoding of vibrissa position during whisking
has yet to be made. The existing data about stimulus encoding by
neurons of the trigeminal ganglion were collected during electrical
stimulations of the motor nerve (Zucker and Welker, 1969) and during
passive mechanical deflections of the vibrissae (Gibson and Welker,
1983; Lichtenstein
et al., 1990; Shoykhet et
al., 2000). In these
data, there are no signs of a labeled-line code as described above.
Yet, a population rate code might be constructed from neurons
exhibiting directional and amplitude dependency (Zucker and Welker,
1969; Gibson
and Welker, 1983; Shoykhet et
al., 2000).
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Sensorimotor Servo Loop |
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), we propose
that the entire sensorimotor loop of the vibrissal system functions
as a servo loop, whose controlled variables are not external, as in
the case of the stick angle in Wiener’s example, but are, rather,
internal. Unlike motor servo control, the purpose of the sensorimotor
servo control is to optimize sensory computation, much like automatic
gain control loops optimize decoding in radio receivers. Several
schemes of sensorimotor servo loops could accomplish optimization
of sensory processing. We delineate here two such schemes.
The first servo scheme is aimed at stabilizing the whisking
frequency in the presence of contact of the vibrissae with objects.
As demonstrated by the data in Figure 3, the whisking
frequency is stable during each whisking bout. A stable whisking
frequency facilitates phase-sensitive sensory computation and thus
might be actively maintained by a sensory-motor servo loop (Fig.
8a), consisting
of another set of PLLs, implemented across S1, M1 and POm.
Experiments on the sensory response of neurons in vibrissa M1 cortex
in awake animals (Kleinfeld et al., 2002) indicate
that M1 neurons compute the fundamental frequency of a complex,
repetitive input. The interpretation of this and related results
could be that vibrissa S1 cortex and M1 cortex, together with
the POm, are part of a PLL that extracts the fundamental frequency
of the input and maintains a stable frequency of whisking.
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). Naturally, the optimal
set-point for a given loop is located at the center of its working
range, since in that condition the loop can handle the largest
input modulations (Ahissar, 1998). Thus, if
vibrissal temporal decoding were accomplished by a single
thalamocortical PLL, the role of the sensorimotor servo loop would be
to maintain the set-point of this PLL at the center of its working
range. Since the set-point of the PLL is determined by the average
input frequency (Ahissar, 1998), maintaining
the optimal setpoint is accomplished by controlling whisking
frequency. The ‘error signal’, whose value determines the motor
output, should be one of the variables composing the PLL’s set-point,
namely spike-count or latency of thalamic or cortical neurons. Of the
two, the spike-count variable is more reliable (Ahissar et
al., 2001a; Sosnik
et al., 2001) and is
probably more suitable for controlling motor activity (Georgopoulos,
1986; Wessberg et
al., 2000).
In the real brain, closed-loop optimization is complicated by
processing in parallel channels. According to the results described
above, temporal decoding in the paralemniscal system is accomplished
by sets of many parallel thalamocortical PLLs, each possessing
a different working range. Although the working range of any
single PLL is limited (Ahissar, 1998), a collection
of PLLs, each having a slightly different working range, can decode
the entire required frequency range. However, such a scheme
poses serious challenges to the system. For example, for a given
input frequency, different PLLs will produce different output
values, depending on their Tc. Thus, for the same
input, some pools of PLLs will produce meaningful outputs while
others, which will be driven out of their working ranges, will
produce nonsense outputs. How can the readout circuit isolate the
‘proper’ PLLs, namely those PLLs whose output is relevant for
sensory computation during the performance of a given task? A
solution to this problem might be ‘built-in’ the sensorimotor
servo loop scheme. By maintaining the whisking frequency centered
on a given frequency, the servo loop in fact selects a range of
PLLs whose optimal set-points, i.e. centers of their working ranges,
are at that frequency. These ‘relevant’ PLLs will present full range
modulations of their output values while other, ‘non-relevant’, PLLs
will exhibit limited modulations due to saturation. Both readout
circuits and motor control circuits should be able to tune to the
PLLs that exhibit the largest modulations. By this tuning, the
sensorimotor servo loop converges to its own set-point, which is thus
determined by the whisking frequency and the profile of activity
modulations across thalamocortical PLLs.
The set-point of the sensorimotor loop depends on the task in hand. For example, object localization, which involves low spatial (and thus also temporal) frequencies, should lead to set-points optimal for low-frequency PLLs. In contrast, fine texture analysis, which usually involves high spatial and temporal frequencies, should lead to set-points optimal for high-frequency PLLs.
In the above two examples, the motor variable used by the servo
loop was the whisking frequency. This is not the only available
variable for servo control. Other variables, such as protraction
velocity or amplitude, can be used as well. For example, when
scanning a textured surface, protraction velocity (V) might
be controlled to optimize the temporal frequency (f) and
compensate for changes in the texture’s spatial frequency (SF),
since f = |V| * SF. It would probably be reasonable to
assume that whisking frequency is usually the variable used to
optimize processing by paralemniscal PLLs, whereas protraction
velocity is used to optimize processing by lemniscal PLLs, if they
exist. This is because paralemniscal PLLs are tuned to whisking
frequencies, which peak near 10 Hz (Fig. 3d), whereas lemniscal PLLs
are probably tuned to higher frequencies, which are produced
while scanning textures and determined by protraction velocity.
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Predictions for the Servo Loop |
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Another prediction of the sensorimotor servo loop is that if sensory information is removed, by some lesion that ‘opens the sensorimotor loop’, the motor system will not be able to maintain a constant whisking profile against changing conditions, such as changes in air pressure or object profiles. Moreover, under these conditions, the motor system might drive the vibrissae to whisk at one of the extreme states: either with maximal or with minimal possible frequency.
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Concluding Remarks |
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).
Neuronal processes that are implemented by single neurons, such as
transduction, conduction, filtering and integration, have been
described through the years. Processes implemented by neuronal
circuits have also been described. Among these are feedforward
transformations and recurrent relaxation by neural networks,
oscillations by excitatory–inhibitory loops, and motor control
by closed loops. To this repertoire we add here sensory computation
by closed loops. We suggest that a significant portion of sensory
processing is implemented by closed-loop computations. We have shown
here how thalamocortical phaselocked loops may decode information
that is encoded in time by the rat vibrissae and how such loops could
be nested within a larger-scale sensory-motor servo loop. Similar
phase-locked loops might operate in the visual (Ahissar and Arieli,
2001) and auditory
(Ahissar et al., 2001b) systems to
decode temporally encoded information.
Much of our understanding of the operation of neural networks comes from physics. Similarly, understanding closed-loop computation should benefit from engineering. Engineers discovered that closed loops often provide extremely elegant solutions for ‘real-world’ problems, the same kind of problems with which living brains are routinely challenged. Yet, engineered closed loops usually lack one important feature, which is inherent in brains: built-in plasticity. It is possible that closedloop computation and plasticity are two of the most critical features which make brains so efficient. Achieving an understanding of the interplay between closed-loop computations and plasticity is a further challenge.
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Footnotes |
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Abstract
Introduction
, of the torques produced by
the synergistic and antagonistic muscles. (c) Functional
scheme for possible feedback control of rhythmic output. This scheme
illustrates the architecture of a phase-locked loop for rhythmic
output generation. An input-controlled internal oscillator, denoted
by ‘~’, is locked to the desired frequency, f0.
Elements of the circuit are labeled with their function, as in part
(a), with the addition that ‘X’ refers to the multiplication
or mixing of two signals and the boxed Bode plot refers to the
low-pass filtering of the signal. The basic forms of the sinusoidal
signals are further indicated. At lock, the actual frequency,
f, equals the desired frequency, f0, and
the input to the controlled oscillator is a constant drive term
equal to Gcos
