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s by action potentials in a mechanosensory neuron of the cockroach
Neuroscience Letters 243 (1998) 113–116
Information transmission at 500 bits/s by action potentials in a mechanosensory neuron of the cockroach Andrew S. French*, Pa¨ivi H. Torkkeli Department of Physiology and Biophysics, Dalhousie University, Halifax, Nova Scotia B3H 4H7, Canada Received 27 November 1997; received in revised form 22 January 1998; accepted 22 January 1998
Abstract Action potentials are widely used to transmit information within nervous systems but information encoding and transmission rates by action potentials are poorly understood. In the absence of knowledge about encoding, most previous work has used signal-to-noise ratios to estimate information capacities. We used a mechanosensory neuron to transmit information by a simple encoding scheme that allowed us to measure the transmission rate directly. Using either mechanical or electrical stimulation, information was transmitted at rates up to 500 bits/s, higher than ever reported before for real action potentials. However, the maximum possible message length decreased strongly with transmission rate, from ~infinite at 100 bits/s to ~100 ms at 500 bits/ s, probably due to ionic adaptation processes within the neuronal membrane. 1998 Published by Elsevier Science Ireland Ltd.
Keywords: Information capacity; Mechanoreceptor; Sensory neuron; Action potential encoding
Action potentials allow nervous systems to transmit information over comparatively long distances, but this function has a cost, because the transmitted signal must be encoded and decoded. Discovering how nervous systems encode information into action potentials has been the subject of theoretical and experimental studies [9,10] but has no simple answer. A related question is how fast information can be transmitted by a nerve axon; the transmission rate. This, in turn, depends on the encoding scheme, which is unknown, but a variety of statistical approaches and putative coding schemes have been used to approach the problem [10]. Estimation of information capacity by general information theory [11] has been particularly useful. The central concept is that a transmitting system is limited by its inherent noise level, because only signals that can be distinguished from the noise are meaningful. Therefore, measuring the signal-to-noise level of a system allows calculation of the maximum possible information transmission rate. The signal-to-noise level can be estimated directly, if the noise is random with zero mean [4], or indirectly via the * Corresponding author. Tel.: +1 902 4941302; fax: +1 902 4942050; e-mail: [email protected]
coherence function, which is relatively easy to measure [1]. However, this assumes that the system is linear, which is never true for action potential encoding systems. Estimates of information capacity from signal-to-noise levels have given values of ~300 bits/s for cricket cercal afferent neurons [14] and ~200 bits/s in spider slit sensilla [6], which are much lower than the rates of 1650 and 2240 bits/s reported for graded signals in fly photoreceptors [2] and spider slit sensilla [6], respectively, reflecting the cost of action potential encoding. However, the questions remain whether higher rates could be obtained from better encoding schemes, and whether linear coherence accurately measures the maximum available rates. Here, we used a simple encoding scheme to transmit information through a mechanosensory neuron, using both mechanical and intracellular electrical stimulation. The nature of action potentials suggests encoding information in a binary code, where an action potential represents a ‘1’ and its absence a ‘0’ (Fig. 1). This code requires the receiver to know when an action potential is to be expected, and one way to do this is to send, or not send, action potentials regularly at a known rate. This is the principle of serial communication, which is widely used for transmitting binary data between computer systems. This
0304-3940/98/$19.00 1998 Published by Elsevier Science Ireland Ltd. All rights reserved PII S0304- 3940(98) 00110- 4
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scheme of neural encoding was originally suggested by MacKay and McCulloch [7], who pointed out that the amount of information transmitted depends on the probability of receiving any particular binary sequence. In the simplest case, which we used here, action potentials have an equal probability of occurring or not occurring in each time division, and occur randomly, so that any pattern of ‘1’ s and ‘0’ s is equally probable. Pseudo-random binary sequences were generated by a shift register [8] and used to deflect the femoral tactile spine of a cockroach or to inject electric current intracellularly into the soma of the tactile spine neuron (Fig. 2). Action potentials were detected in nerve five by two insect pins passed through the femur. Stimuli were produced by a personal computer via a 12-bit digital to analog convertor and action potentials were detected by the same computer with a resolution of 200 ms. For each experiment, the amplitude and duration of the stimulus (deflection or current pulse) was adjusted to give exactly one action potential per stimulus, so that the presence or absence of an action potential within each interval corresponded to one bit of information. This binary signal was compared to the original transmission by an algorithm that reported any missing or excess action potentials. In addition, all signals were displayed on a graphics screen and inspected visually. At rates up to ~100 bits/s, perfect sequences could be transmitted continuously. We used a maximum sequence duration of 10 s for practical reasons, but much longer sequences were possible. At higher rates, two types of errors appeared: missing action potentials and extra action potentials. The latter were always pairs of action potentials close together, rather than occurring during intervals when there should have been silence. However, we only accepted sequences that were perfect, with each time interval containing one or no action potentials, as in the original sequence (Fig. 3).
Fig. 1. Information was transmitted by a simple binary code with time divided into equal intervals, each containing 1 or 0. The upper cartoon shows a random binary sequence converted by a piezoelectric stimulator into 0.7 mm amplitude deflections of a cockroach femoral tactile spine (middle trace). Afferent action potentials in the femoral axon were detected extracellularly (lower trace). A message was successfully encoded and transmitted if each deflection produced a single action potential.
Fig. 2. For mechanical experiments a piezoelectric translator (PZT P841.20; Physik Instrumente, Waldbronn, Germany) moved the femoral tactile spine on the metathoracic leg of an adult cockroach (Periplaneta americana) via 1 mm electrode glass and a loop of silver wire filled with wax (left). A strain-gauge transducer measured spine position and the controller was operated in open loop mode to improve its temporal response. For intracellular current stimulation the tactile spine was cut below the surface of a saline bath and a microelectrode lowered through the spine lumen to penetrate the neuronal soma (right). Microelectrodes (30–60 MQ) made by a laser puller (P-2000; Sutter Instrument Company, Novato, CA, USA) from borosilicate glass (Hilgenberg; Malsfeld, Germany) were driven by an SEC-10L amplifier (npi electronic; Tamm, Germany) using discontinuous single-electrode current-clamp. In both experiments, action potentials were detected in nerve five by a pair of insect pins passed through the femur, amplified (Grass P15; Quincy, MA, USA) and discriminated by a Schmidt trigger.
Rates above 100 bits/s were possible if the transmission period was less than 10 s. To find the maximum duration of accurate transmission at a given rate we sent each sequence 10 times, and recorded the duration for which half of the sequences were received perfectly. Fig. 4 shows average data from these experiments. Sequences could be transmitted reliably up to 500 bits/s if the message duration was reduced to ~100 ms. Higher rates (550 and 600 bits/s) were attempted in each experiment, without reliable success. The mechanical stimulator had finite rise and decay times (Fig. 3), raising the possibility that it limited the transmission rate. To test this possibility we stimulated the neuron with electric current via a microelectrode (Fig. 2). Action potentials were recorded and the data analyzed as before. Transmission rates up to 500 bits/s were again achieved (Figs. 3 and 4), but with similar limitations on message duration. These data extend the maximum rates of information transmission by action potentials to a new high value of 500 bits/s. While the encoding scheme was artificial, it provided a direct measurement of actual information transmitted, rather than an estimate based on assumptions about encoding. Another measure that has been used before is the number of bits transmitted per action potential, with more than 3 bits per action potential being recorded [14]. The scheme used here has a constant value of 2 bits per
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Fig. 3. Examples of action potential encoding at 100 and 500 bits/s for mechanical and electrical stimuli. Mechanical stimuli overlapped significantly at 500 bits/s due to the stimulator time constant, while current stimuli were still clearly separated. Smaller amplitude action potentials from other axons were always visible in the extracellular recordings. Tactile spine action potentials were discriminated by amplitude using a Schmidt trigger.
action potential, since the mean action potential rate is always half the rate in bits per second [7]. In the scheme used here, absolute timing of action potentials within each sample interval carried no additional information. Alternative encoding schemes have been described that use the actual times of occurrence, allowing more bits per action potential to be transmitted [7]. The use of such dynamic encoding and decoding schemes would probably yield higher transmission rates and longer message lengths, because the limitations that we found are probably caused by the neuron’s inability to maintain high firing rates. Some caution must be used concerning the stimuli that we used. For both mechanical and electrical stimulation it was necessary to vary the amplitude and duration of the pulse stimulus to obtain perfect transmission at different rates. Therefore, it is possible that other stimulus combinations could give higher rates. There were some restraints on these parameters. For example, a rate of 500 bits/s requires a maximum stimulus duration of 1 ms, if an equal relaxation time is used. We also noted that transmission at high rates was relatively independent of stimulus amplitude. This reflects the high-pass frequency response of the receptor [3], making it sensitive to velocity, which was limited by the stimulator. These experimental factors suggest that our results provide lower limits to the maximum information transmission rates. However, another caveat concerns the binary patterns that we used, which were generated randomly. A 1 s binary sequence at 100 bits/s has 2100 possible values, so that an exhaustive test of all possible sequences would require 2100 s, or ~1022 years. Therefore, it is theoretically possible that many sequences would have confounded the ability of the
neuron to transmit faithfully, making our values too high. However, similar limitations apply to all other previous investigations that have used white noise, or other stochastic stimuli. The question of the length of time for which a neuron can transmit information at a given rate has not been addressed well before, but is clearly important. All previous information capacity measurement have been based on finite experiments, and so have time limits, which are typically a few seconds. For example, measurements in spider slit sensilla were based on stimuli of 8.192 s duration [6] and gave a maximum rate of ~200 bits/s, in reasonable agreement with Fig. 4. The dependence of transmission rate on message duration here is probably due to ionic events in the neuronal
Fig. 4. The message duration that could be transmitted reliably declined with increasing transmission rate. Average data are for five mechanical experiments and three current injection experiments. The mean duration of perfect transmission at 500 bits/s was 113 ± 55 ms (mean ± SD) with mechanical and 75 ± 25 ms with electrical stimulation.
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membrane, which are partly understood in this neuron [12,13]. Similar processes probably occur in other neurons. Fig. 4 indicates that the mechanical components of this sensory system do not limit the dynamic properties of transduction, confirming previous measurements [5]. Mechanical stimuli actually gave higher values of message duration, but this may be artefactual due to neuronal damage by the microelectrode, or the relative locations of the mechanical transduction and the site of action potential initiation. The functional significance of the temporal limitation to high rates of information transmission is not known. Tactile spines probably serve as perimeter detectors in the animal’s normal posture. A high transmission rate for a short time may be the optimal function for this neuron. All estimates of information transmission rates by action potential signals are significantly lower than for graded potentials, but estimates are increasing as more experiments are performed. While the ultimate objective is to understand how information is normally encoded and decoded, estimates using artificial stimulation can provide useful indications of the capacities of real neurons. We hope that the approach described here can be expanded and applied to a wider range of neurons to learn more about the quantitative properties of action potential encoding in nervous systems. Supported by the Medical Research Council of Canada. [1] Bendat, J.S. and Piersol, A.G., Random Data, Analysis and Measurement Procedures, Wiley, New York, 1981. [2] de Ruyter van Steveninck, R.R. and Laughlin, S.B., The rate of information transfer at graded-potential synapses, Nature, 379 (1996) 642–645.
[3] French, A.S., Sensory transduction in an insect mechanoreceptor: linear and nonlinear properties, Biol. Cybern., 38 (1980) 115–123. [4] French, A.S., Phototransduction in the fly compound eye exhibits temporal resonances and a pure time delay, Nature, 283 (1980) 200–202. [5] French, A.S., The dynamic properties of the action potential encoder in an insect mechanosensory neuron, Biophys. J., 46 (1984) 285–290. [6] Juusola, M. and French, A.S., The efficiency of sensory information coding by mechanoreceptor neurons, Neuron, 18 (1997) 959–968. [7] MacKay, D. and McCulloch, W.S., The limiting information capacity of a neuronal link, Bull. Math. Biophys., 14 (1952) 127–135. [8] Marmarelis, P.Z. and Marmarelis, V.Z., Analysis of Physiological Systems: the White Noise Approach, Plenum Press, New York, 1978. [9] Perkel, D.H. and Bullock, T.H., Neural coding, Neurosci. Res. Program Bull., 6 (1968) 221–348. [10] Rieke, F., Warland. D., de Ruyter van Steveninck, R.R. and Bialek, W., Spikes, Exploring the Neural Code, MIT Press, Cambridge, MA, 1997. [11] Shannon, C.E. and Weaver, W., The Mathematical Theory of Communication, University of Illinois Press, Urbana, IL, 1949. [12] Torkkeli, P.H. and French, A.S., Characterization of a transient outward current in a rapidly adapting insect mechanosensory neuron, Pflu¨gers Arch., 429 (1994) 72–78. [13] Torkkeli, P.H. and French, A.S., Slowly inactivating outward currents in a cuticular mechanoreceptor neuron of the cockroach (Periplaneta americana), J. Neurophysiol., 74 (1995) 1200–1211. [14] Warland, D., Landolfa, M., Miller, J.P. and Bialek, W., Reading between the spikes in the cercal filiform hair receptors of the cricket. In F. Eckman, (Ed.), Analysis and Modeling of Neural Systems, Kluwer Academic, Boston, MA, 1992, pp. 327–333.
Information transmission at 500 bits/s by action potentials in a mechanosensory neuron of the cockroach Andrew S. French*, Pa¨ivi H. Torkkeli Department of Physiology and Biophysics, Dalhousie University, Halifax, Nova Scotia B3H 4H7, Canada Received 27 November 1997; received in revised form 22 January 1998; accepted 22 January 1998
Abstract Action potentials are widely used to transmit information within nervous systems but information encoding and transmission rates by action potentials are poorly understood. In the absence of knowledge about encoding, most previous work has used signal-to-noise ratios to estimate information capacities. We used a mechanosensory neuron to transmit information by a simple encoding scheme that allowed us to measure the transmission rate directly. Using either mechanical or electrical stimulation, information was transmitted at rates up to 500 bits/s, higher than ever reported before for real action potentials. However, the maximum possible message length decreased strongly with transmission rate, from ~infinite at 100 bits/s to ~100 ms at 500 bits/ s, probably due to ionic adaptation processes within the neuronal membrane. 1998 Published by Elsevier Science Ireland Ltd.
Keywords: Information capacity; Mechanoreceptor; Sensory neuron; Action potential encoding
Action potentials allow nervous systems to transmit information over comparatively long distances, but this function has a cost, because the transmitted signal must be encoded and decoded. Discovering how nervous systems encode information into action potentials has been the subject of theoretical and experimental studies [9,10] but has no simple answer. A related question is how fast information can be transmitted by a nerve axon; the transmission rate. This, in turn, depends on the encoding scheme, which is unknown, but a variety of statistical approaches and putative coding schemes have been used to approach the problem [10]. Estimation of information capacity by general information theory [11] has been particularly useful. The central concept is that a transmitting system is limited by its inherent noise level, because only signals that can be distinguished from the noise are meaningful. Therefore, measuring the signal-to-noise level of a system allows calculation of the maximum possible information transmission rate. The signal-to-noise level can be estimated directly, if the noise is random with zero mean [4], or indirectly via the * Corresponding author. Tel.: +1 902 4941302; fax: +1 902 4942050; e-mail: [email protected]
coherence function, which is relatively easy to measure [1]. However, this assumes that the system is linear, which is never true for action potential encoding systems. Estimates of information capacity from signal-to-noise levels have given values of ~300 bits/s for cricket cercal afferent neurons [14] and ~200 bits/s in spider slit sensilla [6], which are much lower than the rates of 1650 and 2240 bits/s reported for graded signals in fly photoreceptors [2] and spider slit sensilla [6], respectively, reflecting the cost of action potential encoding. However, the questions remain whether higher rates could be obtained from better encoding schemes, and whether linear coherence accurately measures the maximum available rates. Here, we used a simple encoding scheme to transmit information through a mechanosensory neuron, using both mechanical and intracellular electrical stimulation. The nature of action potentials suggests encoding information in a binary code, where an action potential represents a ‘1’ and its absence a ‘0’ (Fig. 1). This code requires the receiver to know when an action potential is to be expected, and one way to do this is to send, or not send, action potentials regularly at a known rate. This is the principle of serial communication, which is widely used for transmitting binary data between computer systems. This
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scheme of neural encoding was originally suggested by MacKay and McCulloch [7], who pointed out that the amount of information transmitted depends on the probability of receiving any particular binary sequence. In the simplest case, which we used here, action potentials have an equal probability of occurring or not occurring in each time division, and occur randomly, so that any pattern of ‘1’ s and ‘0’ s is equally probable. Pseudo-random binary sequences were generated by a shift register [8] and used to deflect the femoral tactile spine of a cockroach or to inject electric current intracellularly into the soma of the tactile spine neuron (Fig. 2). Action potentials were detected in nerve five by two insect pins passed through the femur. Stimuli were produced by a personal computer via a 12-bit digital to analog convertor and action potentials were detected by the same computer with a resolution of 200 ms. For each experiment, the amplitude and duration of the stimulus (deflection or current pulse) was adjusted to give exactly one action potential per stimulus, so that the presence or absence of an action potential within each interval corresponded to one bit of information. This binary signal was compared to the original transmission by an algorithm that reported any missing or excess action potentials. In addition, all signals were displayed on a graphics screen and inspected visually. At rates up to ~100 bits/s, perfect sequences could be transmitted continuously. We used a maximum sequence duration of 10 s for practical reasons, but much longer sequences were possible. At higher rates, two types of errors appeared: missing action potentials and extra action potentials. The latter were always pairs of action potentials close together, rather than occurring during intervals when there should have been silence. However, we only accepted sequences that were perfect, with each time interval containing one or no action potentials, as in the original sequence (Fig. 3).
Fig. 1. Information was transmitted by a simple binary code with time divided into equal intervals, each containing 1 or 0. The upper cartoon shows a random binary sequence converted by a piezoelectric stimulator into 0.7 mm amplitude deflections of a cockroach femoral tactile spine (middle trace). Afferent action potentials in the femoral axon were detected extracellularly (lower trace). A message was successfully encoded and transmitted if each deflection produced a single action potential.
Fig. 2. For mechanical experiments a piezoelectric translator (PZT P841.20; Physik Instrumente, Waldbronn, Germany) moved the femoral tactile spine on the metathoracic leg of an adult cockroach (Periplaneta americana) via 1 mm electrode glass and a loop of silver wire filled with wax (left). A strain-gauge transducer measured spine position and the controller was operated in open loop mode to improve its temporal response. For intracellular current stimulation the tactile spine was cut below the surface of a saline bath and a microelectrode lowered through the spine lumen to penetrate the neuronal soma (right). Microelectrodes (30–60 MQ) made by a laser puller (P-2000; Sutter Instrument Company, Novato, CA, USA) from borosilicate glass (Hilgenberg; Malsfeld, Germany) were driven by an SEC-10L amplifier (npi electronic; Tamm, Germany) using discontinuous single-electrode current-clamp. In both experiments, action potentials were detected in nerve five by a pair of insect pins passed through the femur, amplified (Grass P15; Quincy, MA, USA) and discriminated by a Schmidt trigger.
Rates above 100 bits/s were possible if the transmission period was less than 10 s. To find the maximum duration of accurate transmission at a given rate we sent each sequence 10 times, and recorded the duration for which half of the sequences were received perfectly. Fig. 4 shows average data from these experiments. Sequences could be transmitted reliably up to 500 bits/s if the message duration was reduced to ~100 ms. Higher rates (550 and 600 bits/s) were attempted in each experiment, without reliable success. The mechanical stimulator had finite rise and decay times (Fig. 3), raising the possibility that it limited the transmission rate. To test this possibility we stimulated the neuron with electric current via a microelectrode (Fig. 2). Action potentials were recorded and the data analyzed as before. Transmission rates up to 500 bits/s were again achieved (Figs. 3 and 4), but with similar limitations on message duration. These data extend the maximum rates of information transmission by action potentials to a new high value of 500 bits/s. While the encoding scheme was artificial, it provided a direct measurement of actual information transmitted, rather than an estimate based on assumptions about encoding. Another measure that has been used before is the number of bits transmitted per action potential, with more than 3 bits per action potential being recorded [14]. The scheme used here has a constant value of 2 bits per
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Fig. 3. Examples of action potential encoding at 100 and 500 bits/s for mechanical and electrical stimuli. Mechanical stimuli overlapped significantly at 500 bits/s due to the stimulator time constant, while current stimuli were still clearly separated. Smaller amplitude action potentials from other axons were always visible in the extracellular recordings. Tactile spine action potentials were discriminated by amplitude using a Schmidt trigger.
action potential, since the mean action potential rate is always half the rate in bits per second [7]. In the scheme used here, absolute timing of action potentials within each sample interval carried no additional information. Alternative encoding schemes have been described that use the actual times of occurrence, allowing more bits per action potential to be transmitted [7]. The use of such dynamic encoding and decoding schemes would probably yield higher transmission rates and longer message lengths, because the limitations that we found are probably caused by the neuron’s inability to maintain high firing rates. Some caution must be used concerning the stimuli that we used. For both mechanical and electrical stimulation it was necessary to vary the amplitude and duration of the pulse stimulus to obtain perfect transmission at different rates. Therefore, it is possible that other stimulus combinations could give higher rates. There were some restraints on these parameters. For example, a rate of 500 bits/s requires a maximum stimulus duration of 1 ms, if an equal relaxation time is used. We also noted that transmission at high rates was relatively independent of stimulus amplitude. This reflects the high-pass frequency response of the receptor [3], making it sensitive to velocity, which was limited by the stimulator. These experimental factors suggest that our results provide lower limits to the maximum information transmission rates. However, another caveat concerns the binary patterns that we used, which were generated randomly. A 1 s binary sequence at 100 bits/s has 2100 possible values, so that an exhaustive test of all possible sequences would require 2100 s, or ~1022 years. Therefore, it is theoretically possible that many sequences would have confounded the ability of the
neuron to transmit faithfully, making our values too high. However, similar limitations apply to all other previous investigations that have used white noise, or other stochastic stimuli. The question of the length of time for which a neuron can transmit information at a given rate has not been addressed well before, but is clearly important. All previous information capacity measurement have been based on finite experiments, and so have time limits, which are typically a few seconds. For example, measurements in spider slit sensilla were based on stimuli of 8.192 s duration [6] and gave a maximum rate of ~200 bits/s, in reasonable agreement with Fig. 4. The dependence of transmission rate on message duration here is probably due to ionic events in the neuronal
Fig. 4. The message duration that could be transmitted reliably declined with increasing transmission rate. Average data are for five mechanical experiments and three current injection experiments. The mean duration of perfect transmission at 500 bits/s was 113 ± 55 ms (mean ± SD) with mechanical and 75 ± 25 ms with electrical stimulation.
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membrane, which are partly understood in this neuron [12,13]. Similar processes probably occur in other neurons. Fig. 4 indicates that the mechanical components of this sensory system do not limit the dynamic properties of transduction, confirming previous measurements [5]. Mechanical stimuli actually gave higher values of message duration, but this may be artefactual due to neuronal damage by the microelectrode, or the relative locations of the mechanical transduction and the site of action potential initiation. The functional significance of the temporal limitation to high rates of information transmission is not known. Tactile spines probably serve as perimeter detectors in the animal’s normal posture. A high transmission rate for a short time may be the optimal function for this neuron. All estimates of information transmission rates by action potential signals are significantly lower than for graded potentials, but estimates are increasing as more experiments are performed. While the ultimate objective is to understand how information is normally encoded and decoded, estimates using artificial stimulation can provide useful indications of the capacities of real neurons. We hope that the approach described here can be expanded and applied to a wider range of neurons to learn more about the quantitative properties of action potential encoding in nervous systems. Supported by the Medical Research Council of Canada. [1] Bendat, J.S. and Piersol, A.G., Random Data, Analysis and Measurement Procedures, Wiley, New York, 1981. [2] de Ruyter van Steveninck, R.R. and Laughlin, S.B., The rate of information transfer at graded-potential synapses, Nature, 379 (1996) 642–645.
[3] French, A.S., Sensory transduction in an insect mechanoreceptor: linear and nonlinear properties, Biol. Cybern., 38 (1980) 115–123. [4] French, A.S., Phototransduction in the fly compound eye exhibits temporal resonances and a pure time delay, Nature, 283 (1980) 200–202. [5] French, A.S., The dynamic properties of the action potential encoder in an insect mechanosensory neuron, Biophys. J., 46 (1984) 285–290. [6] Juusola, M. and French, A.S., The efficiency of sensory information coding by mechanoreceptor neurons, Neuron, 18 (1997) 959–968. [7] MacKay, D. and McCulloch, W.S., The limiting information capacity of a neuronal link, Bull. Math. Biophys., 14 (1952) 127–135. [8] Marmarelis, P.Z. and Marmarelis, V.Z., Analysis of Physiological Systems: the White Noise Approach, Plenum Press, New York, 1978. [9] Perkel, D.H. and Bullock, T.H., Neural coding, Neurosci. Res. Program Bull., 6 (1968) 221–348. [10] Rieke, F., Warland. D., de Ruyter van Steveninck, R.R. and Bialek, W., Spikes, Exploring the Neural Code, MIT Press, Cambridge, MA, 1997. [11] Shannon, C.E. and Weaver, W., The Mathematical Theory of Communication, University of Illinois Press, Urbana, IL, 1949. [12] Torkkeli, P.H. and French, A.S., Characterization of a transient outward current in a rapidly adapting insect mechanosensory neuron, Pflu¨gers Arch., 429 (1994) 72–78. [13] Torkkeli, P.H. and French, A.S., Slowly inactivating outward currents in a cuticular mechanoreceptor neuron of the cockroach (Periplaneta americana), J. Neurophysiol., 74 (1995) 1200–1211. [14] Warland, D., Landolfa, M., Miller, J.P. and Bialek, W., Reading between the spikes in the cercal filiform hair receptors of the cricket. In F. Eckman, (Ed.), Analysis and Modeling of Neural Systems, Kluwer Academic, Boston, MA, 1992, pp. 327–333.