- Email: [email protected]
A model for the detection of weak ELF electric and magnetic fields
Bioelectrochemistry and Bioenergetics 47 Ž1998. 207–212
A model for the detection of weak ELF electric and magnetic fields Frank S. Barnes
)
UniÕersity of Colorado at Boulder, Department of Electrical and Computer Engineering, Boulder, CO 80309-0425, USA Received 30 January 1998; revised 19 June 1998; accepted 24 September 1998
Abstract A model for the detection of weak electric and magnetic fields is developed by analogy to a phased array antenna and receiver. Pyramidal cells from the cortex of the brain are shown to have elements which can be modeled as an antenna, a mixer amplifier, and a neural network narrow band filter with summing junctions output which could, in turn, modulate the firing rate of a pacemaker cell or ongoing brain oscillations. The signal-to-noise ratio is shown to increase for signals which are coherent in time and space with the square root of the number of elements involved. Additionally, the signal-to-noise ratio may be enhanced by increasing the power spectral density of the ongoing chaotic oscillation at the applied signal frequency. q 1998 Elsevier Science S.A. All rights reserved. Keywords: Electric fields; Cells; Signal-to-noise ratio
1. Introduction In the controversy over the possible biological effects of weak electric and magnetic fields, one of the important issues has been the ability of a cell to sense a signal in the presence of background electrical noise. The problem of the signal-to-noise ratio has been discussed at length in a number of articles w1–4x. These papers show that we would expect important biological processes to have a good signal-to-noise ratio. Additionally, the thermal noise, the 1rf noise and the electrical signals generated by muscle and nerve activity all lead to signals which are large compared to the average electrical signals which we would expect to be generated by environmental fields from power lines, home appliances, etc. w5x. At the same time, sharks, skates and possibly other animals can detect electric potentials as small as 10 nV and electric fields as small as 5 = 10y7 Vrm w6x. A possible explanation of the ability of these animals to sense these low levels of electric fields may lie in the ability of their biological systems to perform coherent signal processing. This subject has been discussed by Mullins et al. w7x. In this paper, we propose to model the biological system by analogy to a phased array antenna and receiver
) Tel.: q1-303-492-8225; [email protected]
fax:
q1-303-492-2758;
e-mail:
system as shown in Fig. 1. Coherent signal processing techniques are used to extract signals from background noise of much larger amplitude in order to obtain radar images, and we wish to show that it may be possible for biological systems to do the same thing. The analogy developed in this paper will be to show that a long, narrow cell behaves like an antenna which concentrates the signal for ELF electrical fields at the membrane. Both stochastic resonance and the neural transmitters at the junction between cells may act to amplify the signal. The outputs from many dendrite junctions may be summed along the core of a pyramidal cell in the cortex of the brain just as they would be in an artificial neural network so as to give a weighted sum of the inputs at the cell body. This weighted sum can then serve as a filter to extract the signal from the background noise. The output from the cell body along an axon may again serve as the input to an oscillator made up of a collection of pyramidal cells in order to modulate the frequency of its firing rate just as it would if the output from a phased array system was used to control the firing rate of a free running multivibrator. In the brain, the oscillators may be formed from a collection of cells which are connected so that a signal can flow from cell to cell and loop back on itself to form an oscillator. The feedback loop parameters are normally such that the system is oscillating in a chaotic mode. To get an estimate of the time scales for these oscillations we can estimate the transit time for signals through
0302-4598r98r$ - see front matter q 1998 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 2 - 4 5 9 8 Ž 9 8 . 0 0 1 9 0 - 1
208
F.S. Barnesr Bioelectrochemistry and Bioenergetics 47 (1998) 207–212
Fig. 1. A possible model for biological detection of weak electric fields.
the pyramidal cells. The velocity of an action potential in brain cells ranges from approximately 1 mrs to 10 mrs w8x with the slow velocities occurring in the small diameter cells. The velocity for a charge wave propagating down the dendrites is estimated to be in the range from 0.1 to 1 mrs w9x. The length of the dendrites is typically less than 1 mm so that the transit time can be estimated to be in the range of two milliseconds. If the velocity for action potential in the axon is approximately 5 mrs and the length is about 5 cm, then the propagation time is on the order of ten milliseconds. The round trip transit time for a loop containing two pyramidal cells with a pulse flowing down one and back on the other is estimated to take on the order of 24 ms, which would lead to an oscillation frequency on the order of 40 Hz. A shorter feedback loop would lead to higher oscillation frequencies. Signals typically travel in parallel along bundles of nerve fibers where many of the nerve fibers are reasonably parallel to each other. Thus an externally applied signal is likely to excite many cells synchronously. Our external signal is a sine wave at either 50 or 60 Hz. Optical signals from computer monitor frame rates have been shown to generate signals in the EEG to more than 70 Hz w10x. Thus we have some evidence that signals in the range of power line frequencies can be detected. Typical membranes have nonlinear, characteristic curves which can partially rectify an externally applied sine wave so that the cell interior hyperpolarizes w11x. The synaptic junctions can be thought of as rectifiers in the sense that they are propagating signals in a given direction. Thus, inputs from an externally applied 60-Hz field might be thought of as generating a periodic series of pulses which are about 8 ms long every 17 ms which can be amplified by the stochastic resonance. It is believed that information is transmitted by controlling the pulse repetition rate. Thus, if this model is validated by experimental studies, we would have a mechanism by which low-level electrical
signals could be used to input information into a biological system as they appear to do in a variety of fish. As time-varying magnetic fields generate electric fields, the model to be presented for the detection of electric fields may also serve as part of a model for the detection of low-level ELF magnetic fields. There are, however, other mechanisms which could serve as direct detectors of magnetic fields. These may involve the existence of chains of magnetite ŽFe 3 O4 . Žiron, oxygen., upon which magnetic fields exert a torque which may allow animals such as bees, homing pigeons and salmon to track the earth’s magnetic fields w12x, or the recombination rates of free radicals w13x. We also have some evidence that more than one mechanism is involved in the effects of low-level magnetic fields on the growth of cells, one of which is effected by the DC magnetic field w14x. Thus it is likely that the model proposed in this paper will only be a partial explanation for some of the possible effects of low-level magnetic fields on people. A variation of this model may apply to modulated radio or microwave signals such as are generated by handheld cell phones. If the thermal time constant for the region of interest lies between the RF carrier frequency and the modulation frequency, even a very small amount of heating may serve to generate a chemical signal with properties similar to those generated by low frequency electric fields w15x.
2. A phased array model for the detection of low-level electrical signals A sketch of the overall model for a brain cell and a phased array antenna and receiver are shown in parallel in Fig. 1. In this model we assume the following. 1. Long thin bodies of a cell act as antennas for the detection of an electric field and concentrate the voltage drop across the membranes.
F.S. Barnesr Bioelectrochemistry and Bioenergetics 47 (1998) 207–212
2. The electrical fields and concentration gradients in the interior of the cell move charge from synaptic junctions in the dendrites to the soma and along the axon to the synaptic junctions at the distal end of the cell, which in turn may activate Naq, Kq and Ca2q channels and release neural transmitters. These neural transmitters may release from 2000 to 10,000 Ca2q ions which serve to amplify the signal. The current flow across the junction or the strength of the connections and the amplifier gain depends on the stimulation history. 3. The charge outputs from the synaptic junctions at the dendrites diffuse down the dendritic chain into the soma. These charges are summed in a graded way so that the input of a coherent signal in time and space from a large number of dendrites is summed over a period of time which is short compared to the time constant for charge to leak out of the cell. This results in a signal at the cell body with a signal-to-noise ratio which is larger than the initial signal to each dendritic junction. 4. Multiple axon inputs to the oscillating system are assumed to be sufficient to change the firing rate, and thus transmit information to other parts of the brain. 5. Stochastic resonate behavior amplifies the low-level coherent signal and increases the signal-to-noise ratio. Let us examine each part of this system in order. First, we have a long thin cell body that is immersed in an electric field, E, as shown in Fig. 2. In the simplest approximation the voltage across the membrane, Vm , is given by Vm s ELr2
Ž 1.
where L is the length of the cell. A better approximation can be obtained from cable theory w16x. In effect, a long cell amplifies the electric field in the membrane by approximately half the ratio of the cell length to the membrane thickness. Cell membranes thicknesses typically are in the range of 5 nm to 10 nm and the cell lengths can range from a few microns to several centimeters. Thus, field strength multiplications of a thousand to about a million at a cell membrane over the field in the interstitial fluid are to be expected. As an example, the peak electric fields induced in a man by a magnetic field of 1 mT at 60 Hz are estimated to be on the order of 10y4 Vrm w17x which would lead to a voltage drop across the membrane of 5 = 10y8 V for a cell
Fig. 2. A long cell of length L.
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a millimeter long. Calculations for a single cylindrical cell 1 mm long and a micron in radius predicts a minimum detectable field of Emin s 2'2
r kTdD f p
1r2
1 3
rL ft
1r2
Ž 2.
where k is Boltzman’s constant, T is the absolute temperature, d is the membrane thickness, L is the cell body length, r is the radius, f is the frequency, r is the membrane resistivity, D f is the bandwidth, and t is the averaging time w3x. Putting in some numbers, an estimated averaging time to detect a field of 10y4 Vrm is about three minutes. If L is increased to 2.5 cm this minimum detectable field decreases by about 125 to about 3 s. This minimum field is larger if 1rf noise is considered in place of the thermal noise as the limiting noise process. However, our pyramidal cell is considerably more complex than a single cylinder Žsee Fig. 3 w18x.. A more realistic estimate of the minimum detectable field would involve a vector sum of the current inputs from the many branches to get an effective area and length of the cell and to get an estimate of the charge accumulation at the interior surface under consideration. If this vector sum were to generate an effective cell surface area with an effective radius of 100 mm, the required averaging time to detect 10y4 Vrm would be reduced to about 16 s, and if the length were extended to include long axon transitions, across a major part of the cortex of several centimeters this time could be reduced by an additional factor on the order of one hundred. A better estimate of this signal could be calculated using a computer simulation program for the pyramidal cell such as GENESIS w19x. Another approach to this problem is to recognize that we are summing a large number of channel currents in parallel—each of which will open at random with a probability that is voltage-dependent. In this situation the noise grows with the square root of the number of channels while the signal grows linearly. Thus, the signal-to-noise ratio grows as the square root of the number of channels being summed in parallel. Take, for example, synaptic junctions in frogs, which typically contain 10 4 to 10 5 channels. Additional synaptic junctions contain vesicles of about 40 nm in diameter that store acetylcholine in rows at the presynaptic terminal. These vesicles contain from 10 3 to 10 4 molecules—and an axon terminal in a muscle cell may have 10 6 or more vesicles. Thus, there are potentially enough events acting in parallel to give us sufficient gain in the signal-to-noise ratio to detect coherent signals well below the thermal noise level as calculated from the membrane resistance and the bandwidth, associated with the typical rise time for a nerve impulse. To reduce the detection time or to reduce the minimum field which can be detected by about a factor of 10 3 would take approximately 10 6 channels operating in parallel, and this approx-
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F.S. Barnesr Bioelectrochemistry and Bioenergetics 47 (1998) 207–212
Fig. 3. Deep pyramidal cell in layer III of pyriform cortex stained by intracellular injection of Lucifer Yellow with antiserum intensification. Fine processes are axon collaterals. Details of distal and proximal apical dendrites are at right. ŽFrom Tseng and Haberly w18x p. 371..
imated the number of Ca2q channels which are estimated to occur in a hippocampal pyramidal cell. It has also been shown that neural transmitters may be released at potentials which are too small to fire an action potential w20x. When these neural transmitters bind to the membrane on the opposite side of a synaptic junction, 2000 to 10,000 Ca2q ions may be released in the interior of the connecting cell from internal stores. The binding of the neural transmitters may last from a few hundred to 10,000 action potentials. The time constants for Ca2q in the proximital dendrites are about 0.6 s, and in the soma, they are about 4 s. These time constants depend on the surface-to-volume ratio. Long-term potentiation, at hippocampal mossy-fiber synapses are shown to follow Hebbian’s Rule w21x. These results indicate that for long-term potentiation to occur, both synaptic activation and postsynaptic depolarization are required. If we again look at Fig. 3 it is clear that the input to each dendrite is the sum of the inputs from many synaptic
junctions and that the soma receives an input which is the weighted sum of the inputs from many dendrites. A single synaptic junction may input enough charge to cause a change in the cell voltage of between 0.5 mV and 1 mV and 10 to 20 synaptic junction inputs are required to fire a nerve cell. Thus the parallel excitation of many dendrites in a bundle would lead to an improved signal-to-noise ratio that grows with the square root of the number of connecting fibers. It has also been observed that it can take from milliseconds to hours for some inhibitions to develop w18x. A lot of work has been done in computer science and electrical engineering to develop artificial neural networks. These networks have been shown to be a very good approach to solving pattern recognition problems. Some years ago we asked ourselves if we could train a computer model of a neural network to identify a 60 Hz sine wave in the presence of noise and if so, how many training runs would it take to identify the presence of the 60 Hz signal with 97% accuracy? This effort showed that we could use
F.S. Barnesr Bioelectrochemistry and Bioenergetics 47 (1998) 207–212
a three-layer neural network with 64 inputs, 8 nodes in the hidden layer and a single output node to recognize a 60 Hz sine wave w22x. We used a backpropagation algorithm and signodal summing at the nodes and showed that the number of training runs required increased as the signal-tonoise ratio decreased. At signal-to-noise ratio of 0.001 it took approximately 1400 cycles to train the network. This would correspond to about 25 s at 60 Hz. It is to be noted that the information presented in the training sequence made a difference and we had to define 59 Hz as a wrong answer if we wanted to separate it from 60 Hz. One way of describing this network is that we trained it to become a narrow band filter with D f - 1 Hz. Thus, it appears that the adaptive properties of synaptic junctions and weighted charge loss as a signal propagates along a dendrite could make it possible to train a pyramidal cell to function like a computer neural network for extracting a signal from noise by narrow banding the response. A better model for a neural network in the brain has been developed by Alkon et al. w23x which would be expected to take fewer cycles to train for a given accuracy. This model requires that signals are carried on collateral fibers repetitively with a temporal relationship in order to modify the network weights to the same postsynaptic junction with a flow-through connection. This kind of excitation is what we would expect to induce from an externally applied field to a nerve bundle. If the output of a number of pyramidal cells is fed, in parallel, to other pyramidal cells in a different part of the cortex in such a way that a signal is fed back to the input cell, the system will form a loop with gain and oscillation. The frequency of the firing of this loop will depend on the length of the path and can be pulled toward that of the incoming signal or locked to it with a signal amplitude which is considerably less than the output signal of the oscillator. The range of frequency pulling and the range for injection locking are determined by the signal-to-noise ratio, the nonlinearity and the loop gain. Moss has shown that stochastic resonate systems provide gain by increasing the frequency component of the output of a chaotic oscillating system at the driving frequency by as much as 30 dB w24x. Thus we may consider a collection of pyramidal cells as having three stages of amplification with different time constants. The first involves release of neural transmitters at the presynaptic junction which bind at the postsynaptic junction and reduce the threshold for firing an action potential. The second involves feedback of neural transmitters which bind to the presynaptic junction, further reducing the voltage or charge required to fire an action potential. The third gain mechanism involves stochastic resonance and increases the frequency component of the normal chaotic oscillations at the input frequency. This differs from the injection locking of coherent oscillations to an external signal in many cases only by small changes in the coefficients of differential equations describing the system. The injection locking of electronic oscillators has been studied
211
for many years and is closely related to the work on phase-locked loops w25,26x. We have shown that pacemaker cells from the ganglion of an Applysia can be phase-locked to an external signal near its resonate frequency with as little as 0.2 nA w2x and external current densities on the order of 0.01 Arm2 w27x. In our electronic model studies we showed that phase-locking could be achieved at signal-to-noise ratios as low as 0.25 near the resonate frequency. The signal required for locking decreased with the reduction of the frequency offset and with increasing signal-to-noise ratios w28x. We would expect closely related behavior from a group of chaotically oscillating cells. In modeling the effects of low-level ELF externally-applied fields, we must presume that the cells are functioning in a normal way during the course of the exposure. This means that the cells are firing with a distribution which may be described as approximately chaotic and where the summation of the fields generated by the cells leads to signals which are closely related to the EEG. Our problem is to determine the minimum electric field which will modify this. The characteristics of stochastic resonate systems has been studied at length by several authors w24,29x. Aslanidi et al. w30x have calculated the effects of Gaussian noise and Gaussian noise plus a sinusoidal signal on the Hodgkin–Huxley model. Their results show that the injection of noise current with an average value of about 20 times the signal leads amplification of the signal by about a factor of 8 and an improvement in the signal-to-noise ratio of about 14. These results are calculated for signal levels which are larger than we would expect Ž15 Arm2 . but demonstrate the significance of the ongoing cell activity and potential for modifying the spiking pattern from the cell with a signal which is smaller than the noise.
3. Conclusion In conclusion, it has been shown that in examining the question of the lowest levels of low-frequency electric and magnetic fields which can be detected by a biological system more than a simple spherical cell needs to be considered. It is likely that the coherence properties of the input signal in both space and time are important. The adaptive properties of neural junctions are likely to make the response of the biological system for repetitive signals over an extended period of time different than for that of short exposures. The chaotic background from ongoing biological activity needs to be included in the system models for low-level detection of electric and magnetic fields. Additionally, for those with an engineering or physical science background, it may be useful to use the techniques which have been developed in signal processing for the extraction of signals from noise as models to suggest possible mechanisms by which biological systems may perform the same function.
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F.S. Barnesr Bioelectrochemistry and Bioenergetics 47 (1998) 207–212
Acknowledgements The author would like to express his appreciation to many colleagues and students for productive discussions which have helped shape his understanding of the complexity of this problem of low-level detection of electric and magnetic fields by biological systems. Of particular importance to this paper have been discussions with H. Wachtel, A. Shepard, I. Davenport, J. Weaver, T. Litovitz, M. Zhadin and the independent study projects by S. Thilenius and A. Smoller.
References w1x F.S. Barnes, M. Seyed-Madani, Some possible limits on the minimum electric signals of biological significance, in: M. Blank, E. Findl ŽEds.., Mechanistic Approaches to Interactions of Electric and Electromagnetic Fields with Living Systems, Plenum, New York, 1987, pp. 339–347. w2x F.S. Barnes, Interaction of DC and ELF electric fields with biological materials and systems, in: C. Polk, E. Postow ŽEds.., The CRC Handbook on Biological Effects of Electromagnetic Fields, CRC Press, Boca Raton, FL, 1996, pp. 103–147. w3x J.C. Weaver, R.D. Astumian, The response of living cells to very weak electric fields: the thermal noise limit, Science 247 Ž1990. 459–461. w4x R.K. Adair, Constraints on biological effects of weak extremelylow-frequency electromagnetic fields, Phys. Rev. A 43 Ž1991. 1039–1048. w5x F.S. Barnes, Typical electric and magnetic field exposures at powerline frequencies and their coupling to biological systems, in: M. Blank ŽEd.., Electromagnetic Fields Biological Interactions and Mechanisms, Advances in Chemistry Series 250, American Chemical Society, Washington, DC, 1995, pp. 37–55. w6x A.J. Kalmijn, Electric and magnetic field detection in elasmobranch fishes, Science 281 Ž1982. 916–918. w7x J.M. Mullins, T.A. Litovitz, G.J. Montrose, The role of coherence in electromagnetic field-Induced bioeffects: the signal-to-noise dilemma, in: M. Blank ŽEd.., Electromagnetic Fields Biological Interactions and Mechanisms, Advances in Chemistry Series 250, American Chemical Society, Washington, DC, 1995, pp. 319–338. w8x M.A.B. Brazier, Electrical Activity of the Nervous System, A Textbook for Students, 3rd edn., Williams and Wilkins, Baltimore, 1968. w9x S. Ochs, Elements of Neurophysiology, Wiley, New York, 1965. w10x Keyl Mild, personal communication. w11x F.S. Barnes, Interaction of DC and ELF electric fields with biological materials and systemsm, in: C. Polk, E. Postow ŽEds.., The CRC Handbook on Biological Effects of Electromagnetic Fields, CRC Press, Boca Raton, FL, 1996. w12x A. Kobayashi, J.L. Kirschvink, Magnetoreception and electromagnetic field effects: sensory perception of the geomagnetic field in animals and humans, in: M. Blank ŽEd.., Electromagnetic Fields Biological Interactions and Mechanisms, Advances in Chemistry Series 250, American Chemical Society, Washington, DC, 1995, pp. 367–394.
w13x J. Walleczek, MagnetoKinetic effects of radical pairs, in: M. Blank ŽEd.., Electromagnetic Fields Biological Interactions and Mechanisms, Advances in Chemistry Series 250, American Chemical Society, Washington, DC, 1995, pp. 395–420. w14x C. Bingham, The Effects of ELF and DC Magnetic Fields on the Division Rate of Mastocytoma Cells, PhD thesis, University of Colorado at Boulder, 1996. w15x F.S. Barnes, Effects of electromagnetic fields on chemical reaction rates, Biofizika ŽRussian Journal of Biophysics. 41 Ž4. Ž1996. 790–797. w16x R.J. MacGregor, E.R. Lewis, Neural modeling: Electrical Signal Processing in the Nervous System, Chap. 6, Plenum Publishing, New York, 1977, pp. 123–152. w17x H. Wachtel, Comparison of endogenous currents in and around cells with those induced by exogenous extremely low frequency magnetic fields, in: M. Blank ŽEd.., Electromagnetic Fields Biological Interactions and Mechanisms, Advances in Chemistry Series 250, American Chemical Society, Washington, DC, 1995, pp. 99–107. w18x G.F. Tseng, L.B. Haberly, Deep neurons in pyriform cortex I, morphology and synaptically evoked responses including a unique high-amplitude paired shock facilitation and deep neurons in pyriform cortex: II. Membrane properties that underlie unusual synaptic responses, J. Neurophysiol. ŽJC7., 62 Ž2. ŽAug 1989. 369-85, 385– 400 ŽFigure on p. 371.. w19x M. Bower, D. Beeman, The Book of GENESIS: Exploring Realistic Neural Models with the GEneral NEural SImulation System, 2nd edn., TELOS w , The Electronic Library of Science, Santa Clara, CA, an imprint of Springer-Verlag New York, 1998. w20x D. Alkon, H. Rasmussen, A spatial-temporal model of cell activation, Science 26 Ž1989. 998–1004. w21x D. Jaffe, D. Johnston, Induction of long-term potentiation at hippocampal mossy-fiber synapses follows a Hebbian Rule, J. Neurophysiol. 64 Ž3. Ž1990. 948–960. w22x F.S. Barnes, Some engineering models for interactions of electric and magnetic fields with biological systems, Bioelectromagnetics Supplement 1 Ž1992. 67–85. w23x D.L. Alkon, K.T. Blackwell, G.S. Barbour, A.K. Rogder, T.P. Vogl, Pattern-Recognition by an Artificial Network Derived from Biological Neuronal Systems, Biological Cybernetics, pp. 841–851. w24x F. Moss, Stochastic resonance, Ber Bansenges Phys. Chem 95 Ž3. Ž1991. 302–311. w25x R. Adler, A study of locking phenomena in oscillators, Proc. Inst. of Radio Engineering ŽIRE. 34 Ž1946. 351. w26x W.C. Lindsey, M.K. Simon, Phase-Locked Loops and Their Application, IEEE Press, New York, 1978. w27x H. Wachtel, Firing-pattern changes and transmembrane currents produced by extremely low frequency fields in pacemaker neurons, in Proc. of the 18th Annual Hanford Life Sciences Symp., Richland, WA, 1978, Technical Information Center, Department of Energy, p. 132. w28x F.S. Barnes, A. Smoller, A. Sheppard, Injection locking of pacemaker cells, in: Proc. of the 9th Annual Conf. IEEE Engineering in Medicine and Biology Society, 1986. w29x B. McNamara, Wiesenfeld, Theory of stochastic resonance, Phys. Rev. A 39 Ž9. Ž1989. 4854–4869. w30x O.V. Aslanidi, A.B. Gapeyeu, V.N. Kazachenko, K.V. Kochetkov, O.A. Mornev, B.F. Santalov, E.E. Fesenko, N.K. Chemeras, Stochastic Resonance in the Hodgkin-Huxley Nerve Fiber Model, 25–28 May, 1997, Pushchino, Proc. of the 1st Intl. Symp. on Elec. Activity of the Brain: Mathematical Models and Analytical Methods, p. 63.
A model for the detection of weak ELF electric and magnetic fields Frank S. Barnes
)
UniÕersity of Colorado at Boulder, Department of Electrical and Computer Engineering, Boulder, CO 80309-0425, USA Received 30 January 1998; revised 19 June 1998; accepted 24 September 1998
Abstract A model for the detection of weak electric and magnetic fields is developed by analogy to a phased array antenna and receiver. Pyramidal cells from the cortex of the brain are shown to have elements which can be modeled as an antenna, a mixer amplifier, and a neural network narrow band filter with summing junctions output which could, in turn, modulate the firing rate of a pacemaker cell or ongoing brain oscillations. The signal-to-noise ratio is shown to increase for signals which are coherent in time and space with the square root of the number of elements involved. Additionally, the signal-to-noise ratio may be enhanced by increasing the power spectral density of the ongoing chaotic oscillation at the applied signal frequency. q 1998 Elsevier Science S.A. All rights reserved. Keywords: Electric fields; Cells; Signal-to-noise ratio
1. Introduction In the controversy over the possible biological effects of weak electric and magnetic fields, one of the important issues has been the ability of a cell to sense a signal in the presence of background electrical noise. The problem of the signal-to-noise ratio has been discussed at length in a number of articles w1–4x. These papers show that we would expect important biological processes to have a good signal-to-noise ratio. Additionally, the thermal noise, the 1rf noise and the electrical signals generated by muscle and nerve activity all lead to signals which are large compared to the average electrical signals which we would expect to be generated by environmental fields from power lines, home appliances, etc. w5x. At the same time, sharks, skates and possibly other animals can detect electric potentials as small as 10 nV and electric fields as small as 5 = 10y7 Vrm w6x. A possible explanation of the ability of these animals to sense these low levels of electric fields may lie in the ability of their biological systems to perform coherent signal processing. This subject has been discussed by Mullins et al. w7x. In this paper, we propose to model the biological system by analogy to a phased array antenna and receiver
) Tel.: q1-303-492-8225; [email protected]
fax:
q1-303-492-2758;
e-mail:
system as shown in Fig. 1. Coherent signal processing techniques are used to extract signals from background noise of much larger amplitude in order to obtain radar images, and we wish to show that it may be possible for biological systems to do the same thing. The analogy developed in this paper will be to show that a long, narrow cell behaves like an antenna which concentrates the signal for ELF electrical fields at the membrane. Both stochastic resonance and the neural transmitters at the junction between cells may act to amplify the signal. The outputs from many dendrite junctions may be summed along the core of a pyramidal cell in the cortex of the brain just as they would be in an artificial neural network so as to give a weighted sum of the inputs at the cell body. This weighted sum can then serve as a filter to extract the signal from the background noise. The output from the cell body along an axon may again serve as the input to an oscillator made up of a collection of pyramidal cells in order to modulate the frequency of its firing rate just as it would if the output from a phased array system was used to control the firing rate of a free running multivibrator. In the brain, the oscillators may be formed from a collection of cells which are connected so that a signal can flow from cell to cell and loop back on itself to form an oscillator. The feedback loop parameters are normally such that the system is oscillating in a chaotic mode. To get an estimate of the time scales for these oscillations we can estimate the transit time for signals through
0302-4598r98r$ - see front matter q 1998 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 2 - 4 5 9 8 Ž 9 8 . 0 0 1 9 0 - 1
208
F.S. Barnesr Bioelectrochemistry and Bioenergetics 47 (1998) 207–212
Fig. 1. A possible model for biological detection of weak electric fields.
the pyramidal cells. The velocity of an action potential in brain cells ranges from approximately 1 mrs to 10 mrs w8x with the slow velocities occurring in the small diameter cells. The velocity for a charge wave propagating down the dendrites is estimated to be in the range from 0.1 to 1 mrs w9x. The length of the dendrites is typically less than 1 mm so that the transit time can be estimated to be in the range of two milliseconds. If the velocity for action potential in the axon is approximately 5 mrs and the length is about 5 cm, then the propagation time is on the order of ten milliseconds. The round trip transit time for a loop containing two pyramidal cells with a pulse flowing down one and back on the other is estimated to take on the order of 24 ms, which would lead to an oscillation frequency on the order of 40 Hz. A shorter feedback loop would lead to higher oscillation frequencies. Signals typically travel in parallel along bundles of nerve fibers where many of the nerve fibers are reasonably parallel to each other. Thus an externally applied signal is likely to excite many cells synchronously. Our external signal is a sine wave at either 50 or 60 Hz. Optical signals from computer monitor frame rates have been shown to generate signals in the EEG to more than 70 Hz w10x. Thus we have some evidence that signals in the range of power line frequencies can be detected. Typical membranes have nonlinear, characteristic curves which can partially rectify an externally applied sine wave so that the cell interior hyperpolarizes w11x. The synaptic junctions can be thought of as rectifiers in the sense that they are propagating signals in a given direction. Thus, inputs from an externally applied 60-Hz field might be thought of as generating a periodic series of pulses which are about 8 ms long every 17 ms which can be amplified by the stochastic resonance. It is believed that information is transmitted by controlling the pulse repetition rate. Thus, if this model is validated by experimental studies, we would have a mechanism by which low-level electrical
signals could be used to input information into a biological system as they appear to do in a variety of fish. As time-varying magnetic fields generate electric fields, the model to be presented for the detection of electric fields may also serve as part of a model for the detection of low-level ELF magnetic fields. There are, however, other mechanisms which could serve as direct detectors of magnetic fields. These may involve the existence of chains of magnetite ŽFe 3 O4 . Žiron, oxygen., upon which magnetic fields exert a torque which may allow animals such as bees, homing pigeons and salmon to track the earth’s magnetic fields w12x, or the recombination rates of free radicals w13x. We also have some evidence that more than one mechanism is involved in the effects of low-level magnetic fields on the growth of cells, one of which is effected by the DC magnetic field w14x. Thus it is likely that the model proposed in this paper will only be a partial explanation for some of the possible effects of low-level magnetic fields on people. A variation of this model may apply to modulated radio or microwave signals such as are generated by handheld cell phones. If the thermal time constant for the region of interest lies between the RF carrier frequency and the modulation frequency, even a very small amount of heating may serve to generate a chemical signal with properties similar to those generated by low frequency electric fields w15x.
2. A phased array model for the detection of low-level electrical signals A sketch of the overall model for a brain cell and a phased array antenna and receiver are shown in parallel in Fig. 1. In this model we assume the following. 1. Long thin bodies of a cell act as antennas for the detection of an electric field and concentrate the voltage drop across the membranes.
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2. The electrical fields and concentration gradients in the interior of the cell move charge from synaptic junctions in the dendrites to the soma and along the axon to the synaptic junctions at the distal end of the cell, which in turn may activate Naq, Kq and Ca2q channels and release neural transmitters. These neural transmitters may release from 2000 to 10,000 Ca2q ions which serve to amplify the signal. The current flow across the junction or the strength of the connections and the amplifier gain depends on the stimulation history. 3. The charge outputs from the synaptic junctions at the dendrites diffuse down the dendritic chain into the soma. These charges are summed in a graded way so that the input of a coherent signal in time and space from a large number of dendrites is summed over a period of time which is short compared to the time constant for charge to leak out of the cell. This results in a signal at the cell body with a signal-to-noise ratio which is larger than the initial signal to each dendritic junction. 4. Multiple axon inputs to the oscillating system are assumed to be sufficient to change the firing rate, and thus transmit information to other parts of the brain. 5. Stochastic resonate behavior amplifies the low-level coherent signal and increases the signal-to-noise ratio. Let us examine each part of this system in order. First, we have a long thin cell body that is immersed in an electric field, E, as shown in Fig. 2. In the simplest approximation the voltage across the membrane, Vm , is given by Vm s ELr2
Ž 1.
where L is the length of the cell. A better approximation can be obtained from cable theory w16x. In effect, a long cell amplifies the electric field in the membrane by approximately half the ratio of the cell length to the membrane thickness. Cell membranes thicknesses typically are in the range of 5 nm to 10 nm and the cell lengths can range from a few microns to several centimeters. Thus, field strength multiplications of a thousand to about a million at a cell membrane over the field in the interstitial fluid are to be expected. As an example, the peak electric fields induced in a man by a magnetic field of 1 mT at 60 Hz are estimated to be on the order of 10y4 Vrm w17x which would lead to a voltage drop across the membrane of 5 = 10y8 V for a cell
Fig. 2. A long cell of length L.
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a millimeter long. Calculations for a single cylindrical cell 1 mm long and a micron in radius predicts a minimum detectable field of Emin s 2'2
r kTdD f p
1r2
1 3
rL ft
1r2
Ž 2.
where k is Boltzman’s constant, T is the absolute temperature, d is the membrane thickness, L is the cell body length, r is the radius, f is the frequency, r is the membrane resistivity, D f is the bandwidth, and t is the averaging time w3x. Putting in some numbers, an estimated averaging time to detect a field of 10y4 Vrm is about three minutes. If L is increased to 2.5 cm this minimum detectable field decreases by about 125 to about 3 s. This minimum field is larger if 1rf noise is considered in place of the thermal noise as the limiting noise process. However, our pyramidal cell is considerably more complex than a single cylinder Žsee Fig. 3 w18x.. A more realistic estimate of the minimum detectable field would involve a vector sum of the current inputs from the many branches to get an effective area and length of the cell and to get an estimate of the charge accumulation at the interior surface under consideration. If this vector sum were to generate an effective cell surface area with an effective radius of 100 mm, the required averaging time to detect 10y4 Vrm would be reduced to about 16 s, and if the length were extended to include long axon transitions, across a major part of the cortex of several centimeters this time could be reduced by an additional factor on the order of one hundred. A better estimate of this signal could be calculated using a computer simulation program for the pyramidal cell such as GENESIS w19x. Another approach to this problem is to recognize that we are summing a large number of channel currents in parallel—each of which will open at random with a probability that is voltage-dependent. In this situation the noise grows with the square root of the number of channels while the signal grows linearly. Thus, the signal-to-noise ratio grows as the square root of the number of channels being summed in parallel. Take, for example, synaptic junctions in frogs, which typically contain 10 4 to 10 5 channels. Additional synaptic junctions contain vesicles of about 40 nm in diameter that store acetylcholine in rows at the presynaptic terminal. These vesicles contain from 10 3 to 10 4 molecules—and an axon terminal in a muscle cell may have 10 6 or more vesicles. Thus, there are potentially enough events acting in parallel to give us sufficient gain in the signal-to-noise ratio to detect coherent signals well below the thermal noise level as calculated from the membrane resistance and the bandwidth, associated with the typical rise time for a nerve impulse. To reduce the detection time or to reduce the minimum field which can be detected by about a factor of 10 3 would take approximately 10 6 channels operating in parallel, and this approx-
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Fig. 3. Deep pyramidal cell in layer III of pyriform cortex stained by intracellular injection of Lucifer Yellow with antiserum intensification. Fine processes are axon collaterals. Details of distal and proximal apical dendrites are at right. ŽFrom Tseng and Haberly w18x p. 371..
imated the number of Ca2q channels which are estimated to occur in a hippocampal pyramidal cell. It has also been shown that neural transmitters may be released at potentials which are too small to fire an action potential w20x. When these neural transmitters bind to the membrane on the opposite side of a synaptic junction, 2000 to 10,000 Ca2q ions may be released in the interior of the connecting cell from internal stores. The binding of the neural transmitters may last from a few hundred to 10,000 action potentials. The time constants for Ca2q in the proximital dendrites are about 0.6 s, and in the soma, they are about 4 s. These time constants depend on the surface-to-volume ratio. Long-term potentiation, at hippocampal mossy-fiber synapses are shown to follow Hebbian’s Rule w21x. These results indicate that for long-term potentiation to occur, both synaptic activation and postsynaptic depolarization are required. If we again look at Fig. 3 it is clear that the input to each dendrite is the sum of the inputs from many synaptic
junctions and that the soma receives an input which is the weighted sum of the inputs from many dendrites. A single synaptic junction may input enough charge to cause a change in the cell voltage of between 0.5 mV and 1 mV and 10 to 20 synaptic junction inputs are required to fire a nerve cell. Thus the parallel excitation of many dendrites in a bundle would lead to an improved signal-to-noise ratio that grows with the square root of the number of connecting fibers. It has also been observed that it can take from milliseconds to hours for some inhibitions to develop w18x. A lot of work has been done in computer science and electrical engineering to develop artificial neural networks. These networks have been shown to be a very good approach to solving pattern recognition problems. Some years ago we asked ourselves if we could train a computer model of a neural network to identify a 60 Hz sine wave in the presence of noise and if so, how many training runs would it take to identify the presence of the 60 Hz signal with 97% accuracy? This effort showed that we could use
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a three-layer neural network with 64 inputs, 8 nodes in the hidden layer and a single output node to recognize a 60 Hz sine wave w22x. We used a backpropagation algorithm and signodal summing at the nodes and showed that the number of training runs required increased as the signal-tonoise ratio decreased. At signal-to-noise ratio of 0.001 it took approximately 1400 cycles to train the network. This would correspond to about 25 s at 60 Hz. It is to be noted that the information presented in the training sequence made a difference and we had to define 59 Hz as a wrong answer if we wanted to separate it from 60 Hz. One way of describing this network is that we trained it to become a narrow band filter with D f - 1 Hz. Thus, it appears that the adaptive properties of synaptic junctions and weighted charge loss as a signal propagates along a dendrite could make it possible to train a pyramidal cell to function like a computer neural network for extracting a signal from noise by narrow banding the response. A better model for a neural network in the brain has been developed by Alkon et al. w23x which would be expected to take fewer cycles to train for a given accuracy. This model requires that signals are carried on collateral fibers repetitively with a temporal relationship in order to modify the network weights to the same postsynaptic junction with a flow-through connection. This kind of excitation is what we would expect to induce from an externally applied field to a nerve bundle. If the output of a number of pyramidal cells is fed, in parallel, to other pyramidal cells in a different part of the cortex in such a way that a signal is fed back to the input cell, the system will form a loop with gain and oscillation. The frequency of the firing of this loop will depend on the length of the path and can be pulled toward that of the incoming signal or locked to it with a signal amplitude which is considerably less than the output signal of the oscillator. The range of frequency pulling and the range for injection locking are determined by the signal-to-noise ratio, the nonlinearity and the loop gain. Moss has shown that stochastic resonate systems provide gain by increasing the frequency component of the output of a chaotic oscillating system at the driving frequency by as much as 30 dB w24x. Thus we may consider a collection of pyramidal cells as having three stages of amplification with different time constants. The first involves release of neural transmitters at the presynaptic junction which bind at the postsynaptic junction and reduce the threshold for firing an action potential. The second involves feedback of neural transmitters which bind to the presynaptic junction, further reducing the voltage or charge required to fire an action potential. The third gain mechanism involves stochastic resonance and increases the frequency component of the normal chaotic oscillations at the input frequency. This differs from the injection locking of coherent oscillations to an external signal in many cases only by small changes in the coefficients of differential equations describing the system. The injection locking of electronic oscillators has been studied
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for many years and is closely related to the work on phase-locked loops w25,26x. We have shown that pacemaker cells from the ganglion of an Applysia can be phase-locked to an external signal near its resonate frequency with as little as 0.2 nA w2x and external current densities on the order of 0.01 Arm2 w27x. In our electronic model studies we showed that phase-locking could be achieved at signal-to-noise ratios as low as 0.25 near the resonate frequency. The signal required for locking decreased with the reduction of the frequency offset and with increasing signal-to-noise ratios w28x. We would expect closely related behavior from a group of chaotically oscillating cells. In modeling the effects of low-level ELF externally-applied fields, we must presume that the cells are functioning in a normal way during the course of the exposure. This means that the cells are firing with a distribution which may be described as approximately chaotic and where the summation of the fields generated by the cells leads to signals which are closely related to the EEG. Our problem is to determine the minimum electric field which will modify this. The characteristics of stochastic resonate systems has been studied at length by several authors w24,29x. Aslanidi et al. w30x have calculated the effects of Gaussian noise and Gaussian noise plus a sinusoidal signal on the Hodgkin–Huxley model. Their results show that the injection of noise current with an average value of about 20 times the signal leads amplification of the signal by about a factor of 8 and an improvement in the signal-to-noise ratio of about 14. These results are calculated for signal levels which are larger than we would expect Ž15 Arm2 . but demonstrate the significance of the ongoing cell activity and potential for modifying the spiking pattern from the cell with a signal which is smaller than the noise.
3. Conclusion In conclusion, it has been shown that in examining the question of the lowest levels of low-frequency electric and magnetic fields which can be detected by a biological system more than a simple spherical cell needs to be considered. It is likely that the coherence properties of the input signal in both space and time are important. The adaptive properties of neural junctions are likely to make the response of the biological system for repetitive signals over an extended period of time different than for that of short exposures. The chaotic background from ongoing biological activity needs to be included in the system models for low-level detection of electric and magnetic fields. Additionally, for those with an engineering or physical science background, it may be useful to use the techniques which have been developed in signal processing for the extraction of signals from noise as models to suggest possible mechanisms by which biological systems may perform the same function.
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Acknowledgements The author would like to express his appreciation to many colleagues and students for productive discussions which have helped shape his understanding of the complexity of this problem of low-level detection of electric and magnetic fields by biological systems. Of particular importance to this paper have been discussions with H. Wachtel, A. Shepard, I. Davenport, J. Weaver, T. Litovitz, M. Zhadin and the independent study projects by S. Thilenius and A. Smoller.
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w13x J. Walleczek, MagnetoKinetic effects of radical pairs, in: M. Blank ŽEd.., Electromagnetic Fields Biological Interactions and Mechanisms, Advances in Chemistry Series 250, American Chemical Society, Washington, DC, 1995, pp. 395–420. w14x C. Bingham, The Effects of ELF and DC Magnetic Fields on the Division Rate of Mastocytoma Cells, PhD thesis, University of Colorado at Boulder, 1996. w15x F.S. Barnes, Effects of electromagnetic fields on chemical reaction rates, Biofizika ŽRussian Journal of Biophysics. 41 Ž4. Ž1996. 790–797. w16x R.J. MacGregor, E.R. Lewis, Neural modeling: Electrical Signal Processing in the Nervous System, Chap. 6, Plenum Publishing, New York, 1977, pp. 123–152. w17x H. Wachtel, Comparison of endogenous currents in and around cells with those induced by exogenous extremely low frequency magnetic fields, in: M. Blank ŽEd.., Electromagnetic Fields Biological Interactions and Mechanisms, Advances in Chemistry Series 250, American Chemical Society, Washington, DC, 1995, pp. 99–107. w18x G.F. Tseng, L.B. Haberly, Deep neurons in pyriform cortex I, morphology and synaptically evoked responses including a unique high-amplitude paired shock facilitation and deep neurons in pyriform cortex: II. Membrane properties that underlie unusual synaptic responses, J. Neurophysiol. ŽJC7., 62 Ž2. ŽAug 1989. 369-85, 385– 400 ŽFigure on p. 371.. w19x M. Bower, D. Beeman, The Book of GENESIS: Exploring Realistic Neural Models with the GEneral NEural SImulation System, 2nd edn., TELOS w , The Electronic Library of Science, Santa Clara, CA, an imprint of Springer-Verlag New York, 1998. w20x D. Alkon, H. Rasmussen, A spatial-temporal model of cell activation, Science 26 Ž1989. 998–1004. w21x D. Jaffe, D. Johnston, Induction of long-term potentiation at hippocampal mossy-fiber synapses follows a Hebbian Rule, J. Neurophysiol. 64 Ž3. Ž1990. 948–960. w22x F.S. Barnes, Some engineering models for interactions of electric and magnetic fields with biological systems, Bioelectromagnetics Supplement 1 Ž1992. 67–85. w23x D.L. Alkon, K.T. Blackwell, G.S. Barbour, A.K. Rogder, T.P. Vogl, Pattern-Recognition by an Artificial Network Derived from Biological Neuronal Systems, Biological Cybernetics, pp. 841–851. w24x F. Moss, Stochastic resonance, Ber Bansenges Phys. Chem 95 Ž3. Ž1991. 302–311. w25x R. Adler, A study of locking phenomena in oscillators, Proc. Inst. of Radio Engineering ŽIRE. 34 Ž1946. 351. w26x W.C. Lindsey, M.K. Simon, Phase-Locked Loops and Their Application, IEEE Press, New York, 1978. w27x H. Wachtel, Firing-pattern changes and transmembrane currents produced by extremely low frequency fields in pacemaker neurons, in Proc. of the 18th Annual Hanford Life Sciences Symp., Richland, WA, 1978, Technical Information Center, Department of Energy, p. 132. w28x F.S. Barnes, A. Smoller, A. Sheppard, Injection locking of pacemaker cells, in: Proc. of the 9th Annual Conf. IEEE Engineering in Medicine and Biology Society, 1986. w29x B. McNamara, Wiesenfeld, Theory of stochastic resonance, Phys. Rev. A 39 Ž9. Ž1989. 4854–4869. w30x O.V. Aslanidi, A.B. Gapeyeu, V.N. Kazachenko, K.V. Kochetkov, O.A. Mornev, B.F. Santalov, E.E. Fesenko, N.K. Chemeras, Stochastic Resonance in the Hodgkin-Huxley Nerve Fiber Model, 25–28 May, 1997, Pushchino, Proc. of the 1st Intl. Symp. on Elec. Activity of the Brain: Mathematical Models and Analytical Methods, p. 63.