Finding unstable periodic orbits in electroreceptors, cold receptors and ...
Neurocomputing 26}27 (1999) 79}86
Finding unstable periodic orbits in electroreceptors, cold receptors and hypothalamic neurons夽 Hans A. Braun , Mathias Dewald , Karlheinz Voigt , Martin Huber, Xing Pei, Frank Moss * Institute of Physiology, University of Marburg, D-35037 Marburg, Germany Department of Psychiatry, University of Marburg, D-35037 Marburg, Germany Department of Physics and Astronomy, College of Arts and Science, Center for Neurodynamics, University of Missouri at St. Louis, St. Louis, MO 63121, USA Accepted 18 December 1998
Abstract Recently, searches for low-dimensional dynamical activity in various biological preparations have become fashionable. In particular, demonstrations of the existence of unstable periodic orbits (UPOs) and bifurcations between unstable and stable states are of interest. Here, we report the detection of UPOs in three diverse preparations: electroreceptor, cold receptor and hypothalamic neurons. Inherent, noise mediated oscillators are common to these temperature sensitive neurons, and the UPOs arise in them in the absence of external periodic stimulation. The external temperature is the bifurcation parameter. Behaviors in response to temperature transients are also shown. 1999 Elsevier Science B.V. All rights reserved. Keywords: Unstable periodic orbit; Electroreceptor; Cold receptor; Hypothalamic neuron
1. Introduction In the structural view of dissipative chaos, the dynamics arises from an in"nite set of unstable periodic orbits (UPOs) [4,5]. A chaotic trajectory spends much of its time wandering among the high-order periods, but occasionally visits those of low period.
* Corresponding author. Tel.:#1-314-516-5931; fax: #01-314-516-6152; e-mail: mossf@umslvma. umsl.edu. 夽 Supported in part by the U.S. O$ce of Naval Research, Physics Division, U.S. Department of Energy, the European Science Foundation, Volkswagen Stiftung and the Deutsche Forschungsgemeinschaft. 0925-2312/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 2 3 1 2 ( 9 9 ) 0 0 0 0 9 - 0
80
H.A. Braun et al. / Neurocomputing 26}27 (1999) 79}86
In noise free, stationary, physical systems, periods as high as 95 [8] are experimentally detectable. The relatively recent advent of statistically and topologically based search algorithms [11}16,18] have made it possible to observe UPOs with statistical certainty in various biological preparations including electroreceptors [1] cold receptors [2] synaptic currents [7], bursting hippocampal neurons [10,15,17], human electroencephelographic records [9], and human cardiac dynamics [11]. But because biological preparations are typically non-stationary and include relatively large intensities of dynamical noise, only the "rst few periods have been experimentally observed in the biological preparations. Since higher-order periods are in general less likely to occur, we here focus only on the period-one orbits. In the following sections, we describe the "rst observations of UPOs in preparations from three radically di!erent types of temperature sensitive neurons: cat"sh electroreceptors, rat facial cold receptors and hypothalamic neurons from the rat brain paraventricular nucleus. We brie#y describe the data analysis method, techniques for preparing the neurons, experimental protocols and the results. We conclude with a brief summary.
2. Data analysis method All our data are in the form of a series of time intervals T, between neural action potentials. We seek particular sequences of six time intervals interspersed among the rest of the "le. These sequences are the signatures of an encounter of the general trajectory with a period-1 UPO. Beginning the analysis, we plot the data on a "rst return map, ¹ versus ¹ as shown in Fig. 1A. On this plot, a periodic point is one L> L which lies on the 453 line, that is ¹ ,¹ . An encounter with an unstable periodic L> L point is marked by a sequence of points which approach the 453 line with decreasing distances along a direction called the stable manifold followed immediately by a sequence which departs along the unstable manifold with increasing distances. An encounter is de"ned as the occurrence in the data "le of three points which approach
Fig. 1. Cat"sh electroreceptor. (A) Scatter plot of a complete record of time intervals. (B) An example encounter showing sequential points along the stable direction (squares, inward pointing arrows) and unstable direction (circles, outward arrows).
H.A. Braun et al. / Neurocomputing 26}27 (1999) 79}86
81
Fig. 2. Data from the cat"sh electroreceptor showing (top-to-bottom) scatter plots, histogram of encounters, intervals with K-values and "le segments indicated by the bar lengths, "ring rate and temperature time course. Stable and unstable sequences are indicated by squares and circles, respectively, with the mid-point shown by the large diamonds.
followed by three which depart. An example is shown in Fig. 1B. Searching the data "le results in N such occurrences. The points encompassing all identi"ed sequences are plotted over the complete "les in the examples shown in the top panel of Fig. 2. The noise results in the scattering of these collections of points, but in some cases (where
82
H.A. Braun et al. / Neurocomputing 26}27 (1999) 79}86
UPOs are clearly present in signi"cant concentrations) two populations, lying generally along the unstable and stable directions, can easily be identi"ed. The problem is the noise. If the dynamics contains noise, then the de"nition of an encounter can be ful"lled by accident. Therefore the question becomes: Is N a statistically signi"cant number? This question can be answered by comparing N with the average number 1N 2 found using the same encounter de"nition in randomized, or surrogate "les [19]. Proper surrogates will destroy the essential information contained in the de"nition of an encounter, that is, the sequential order of the seven time intervals. Thus all such sequences except random occurrences will be destroyed by randomly scrambling the order of the time intervals in the data "le. To obtain the mean, we scramble each data "le 100 times, "nd 1N 2 and the standard deviation p. We then examine the statistic, N!1N 2 K" p
(1)
in units of the standard deviation, sigma. K53 sigmas signi"es the detection of UPOs in the data "le with greater than 99% con"dence. The value of K indicates their concentration. Often statistically signi"cant determinations can be made rapidly from "les of only a few hundred time intervals, thus indicating that this algorithm is suitable for on-line determinations of the dynamical state of a preparation including transient changes as described below. Such algorithmic simplicity, and hence speed, is especially important in applications involving chaos control [6,15,3].
3. Experimental methods In all cases, a!erent action potentials were recorded extracellularly. The data analysis was performed o! line on the time intervals between the fast rises of sequential pairs of action potentials. For the peripheral neurons } electroreceptors and cold receptors } the surface temperature overlying the receptive "eld of the neuron was regulated and could be adjusted on demand either in discontinuous steps or swept over a range from 5 to 353C. In the case of the hypothalamus, the temperature of the #uid in a perfusion chamber containing the brain slice preparation could be adjusted in the same manner. The electroreceptor experiments were performed on live anesthetized cat"sh, Ictaluris nebulosus immobilized in a Lucite holder and arti"cially respired [1]. A small chamber containing thermally regulated water was sealed by O-ring over the receptive "eld. Recording was from the ampullary canal located in the dorsal head region using tungsten microelectrodes and standard extracellular electronics. The cold receptor experiments were performed on anaesthetized rats [2]. Recordings were made from single a!erent "bers of the infraorbital nerve with platinum electrodes. The temperature of the receptive "eld was controlled by a brass cylinder (3 mm diam. tip) over the range described above.
H.A. Braun et al. / Neurocomputing 26}27 (1999) 79}86
83
The hypothalamic recordings were made from brain slice preparations obtained from Sprague-Dawley rats. After decapitation, brain removal and conditioning, a block containing the hypothalamus was cut from the brain. Slices (400 lm thick, 5}7 mm diam.) were obtained and conditioned in oxygenated aCSF solution for at least one hour. The slice preparations were then mounted in a temperature controlled perfusion chamber. Single-unit recordings were made from the paraventricular nuclei (PVN) using glass micropipettes positioned with micromanipulators. File segments containing several hundred to three thousand time intervals were analyzed for the presence of period-one UPOs. As the following data demonstrate, the analysis method described here is suitable for examining the transient appearance and disappearance of UPOs in response to step and/or periodic variations in the temperature of as little as $13C.
4. Results Fig. 2 shows some example results from experiments on the cat"sh electroreceptor, though a more complete account has been made elsewhere [1]. The top panel shows four "rst return maps. The complete data "les from which these maps were constructed are shown by the open diamonds. The selected encounters are shown by enlarged symbols: squares indicating the stable directions and circles the unstable direction. The middle panel shows the temporal record of all time intervals ID (small diamonds), together with the time interval sequences comprising the identi"ed encounters in enlarged symbols (squares, stable; circles, unstable directions; large diamonds, the interval pair closest to the "xed point). The solid bars above show the "le segments analyzed and the respective K-values. The bottom panel shows the temperature time course. Note the response to the step temperature increase (30}353C) at about 85 s. Prior to this change, the value K"1.6 indicates no statistically signi"cant concentration of UPOs. However, the larger K-values after the change as well as the histograms of identi"ed encounters E, plotted above indicate the appearance of large concentrations of UPOs in response to the temperature change. These large concentrations (K"9.9 and 10.2), however, accommodate after about 175 s to insigni"cant values. The return maps in the top panel echo these "ndings: only the center two for the largest K-values show two well-de"ned populations of encounter points. In this case, the appearance of UPOs is clearly a transient phenomenon. We note that the mean "ring rate F, shown in a lower panel, indicates very little sensitivity to the temperature step. Some results of our experiments with the rat facial cold receptor [2] are shown in Fig. 3. Here, we illustrate the response of the neuron to a sinusoidal temperature stimulus of only $13C at an average temperature of 293C. The temporal record of all time intervals ID, is shown with bars and K-values above. The identi"ed encounters are again indicated by enlarged symbols. The histogram E, in the top panel shows that the encounter events indicating large concentrations of UPOs are
84
H.A. Braun et al. / Neurocomputing 26}27 (1999) 79}86
Fig. 3. Rat facial cold receptor showing (top-to-bottom) histogram of encounters, intervals with K-values and bars delineating the analyzed "le lengths, "ring rate and temperature time course.
phase locked to the stimulus sine wave albeit with a phase shift of about 3p/4, that is, the large concentrations appear on the downward phase of the temperature cycle. In this case the "ring rate F, shows a similar behavior though it seems to lag the UPO concentration somewhat. The UPOs enter during the increasing phases of the "ring rate cycle. Fig. 4 shows some results obtained from a hypothalamic neuron in the PVN. In this case, a linearly increasing temperature ramp is applied. Note the bifurcation to bursting beginning at approximately 200 s. In this case the K-values and histogram show that a large concentration of UPOs enters the dynamics at about 110 s after the ramp begins prior to the bursting bifurcation. After the bifurcation the neuron accommodates and the UPOs disappear. The "ring rate shows no indication of this behavior.
5. Summary In summary, we have demonstrated the existence of period-one UPOs in three diverse types of temperature sensitive neurons. In response to step, periodic or ramp temperature stimuli, the UPOs enter and depart transiently.
H.A. Braun et al. / Neurocomputing 26}27 (1999) 79}86
85
Fig. 4. Paraventricular nucleus neuron of rat brain with data arranged as in Fig. 3.
References [1] H.A. Braun, K. SchaK fer, K. Voigt, R. Peters, F. Bretschneider, X. Pei, L. Wilkens, F. Moss, Lowdimensional dynamics in sensory biology 1: thermally sensitive electroreceptors of the cat"sh, J. Comput. Neurosci. 4 (1997) 335}347. [2] H.A. Braun, M. Dewald, K. SchaK fer, K. Voigt, X. Pei, F. Moss, Low-dimensional dynamics in sensory biology 2: facial cold receptors of the rat, J. Comput. Neurosci., in press. [3] D.J. Christini, J.J. Collins, Controlling nonchaotic neuronal noise using chaos control techniques, Phys. Rev. Lett. 75 (1995) 2782}2785. [4] P. Cvitanovic, Invariant measurement of strange sets in terms of cycles, Phys. Rev. Lett. 61 (1988) 2729}2732. [5] P. Cvitanovic, Periodic orbits as the skeleton of classical and quantum chaos, Physica D 51 (1991) 138}152. [6] W.L. Ditto, S.N. Rauseo, M.L. Spano, Experimental control of chaos, Phys. Rev. Lett. 65 (1990) 3211}3214. [7] P. Faure, H. Korn, A nonrandom dynamic component in the synaptic noise of a central neuron, Proc. Natl. Acad. Sci. USA 94 (1997) 6506}6511. [8] E.R. Hunt, Stabilizing high-period orbits in a chaotic system-the diode resonator, Phys. Rev. Lett. 67 (1991) 1953}1955. [9] M. Le Van Quyen, J. Martinerie, C. Adam, F.J. Varela, Unstable periodic orbits in human epileptic activity, Phys. Rev. E. 56 (1997) 3401}3411.
86
H.A. Braun et al. / Neurocomputing 26}27 (1999) 79}86
[10] L. Menendez de la Prida, N. Stollenwerk, J.V. Sanchez-Andres, Bursting as a source for predictability in biological neural networks activity, Physica D 110 (1997) 323}331. [11] K. Narayanan, R.B. Govindan, M.S. Gopinathan, Unstable periodic orbits in human cardiac rhythms, Phys. Rev. E 57 (1998) 4594}4736. [12] X. Pei, F. Moss, Characterization of low dimensional dynamics in the cray"sh caudal photoreceptor, Nature 379 (1996a) 618}621. [13] X. Pei, F. Moss, Detecting low dimensional dynamics in biological experiments, Int. J. Neural Systems 7 (1996b) 429-435. [14] D. Pierson, F. Moss, Detecting periodic unstable points in noisy chaotic and limit cycle attractors with applications to biology, Phys. Rev. Lett. 75 (1995) 2124}2127. [15] S.J. Schi!, K. Jerger, D. Duong, T. Chang, M.L. Spano, W.L. Ditto, Controlling chaos in the brain, Nature 370 (1994) 615}620. [16] P. So, E. Ott, S.J. Schi!, D.T. Kaplan, T. Sauer, C. Grebogi, Detecting unstable periodic orbits in chaotic experimental data, Phys. Rev. Lett. 76 (1996) 4705}4708. [17] P. So, E. Ott, T. Sauer, B.J. Gluckman, C. Grebogi, S.J. Schi!, Extracting unstable periodic orbits from chaotic time series data, Phys. Rev. E 55 (1997) 5398}5417. [18] P. So, J.T. Francis, T.I. Neto!, B.J. Gluckman, S.J. Schi!, Periodic orbits: a new language for neuronal dynamics, Biophys. J. 74 (1998) 2776}2785. [19] J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, J.D. Farmer, Testing for nonlinearity in time series: the method of surrogate data, Physica D 58 (1992) 77}94.
Finding unstable periodic orbits in electroreceptors, cold receptors and hypothalamic neurons夽 Hans A. Braun , Mathias Dewald , Karlheinz Voigt , Martin Huber, Xing Pei, Frank Moss * Institute of Physiology, University of Marburg, D-35037 Marburg, Germany Department of Psychiatry, University of Marburg, D-35037 Marburg, Germany Department of Physics and Astronomy, College of Arts and Science, Center for Neurodynamics, University of Missouri at St. Louis, St. Louis, MO 63121, USA Accepted 18 December 1998
Abstract Recently, searches for low-dimensional dynamical activity in various biological preparations have become fashionable. In particular, demonstrations of the existence of unstable periodic orbits (UPOs) and bifurcations between unstable and stable states are of interest. Here, we report the detection of UPOs in three diverse preparations: electroreceptor, cold receptor and hypothalamic neurons. Inherent, noise mediated oscillators are common to these temperature sensitive neurons, and the UPOs arise in them in the absence of external periodic stimulation. The external temperature is the bifurcation parameter. Behaviors in response to temperature transients are also shown. 1999 Elsevier Science B.V. All rights reserved. Keywords: Unstable periodic orbit; Electroreceptor; Cold receptor; Hypothalamic neuron
1. Introduction In the structural view of dissipative chaos, the dynamics arises from an in"nite set of unstable periodic orbits (UPOs) [4,5]. A chaotic trajectory spends much of its time wandering among the high-order periods, but occasionally visits those of low period.
* Corresponding author. Tel.:#1-314-516-5931; fax: #01-314-516-6152; e-mail: mossf@umslvma. umsl.edu. 夽 Supported in part by the U.S. O$ce of Naval Research, Physics Division, U.S. Department of Energy, the European Science Foundation, Volkswagen Stiftung and the Deutsche Forschungsgemeinschaft. 0925-2312/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 2 3 1 2 ( 9 9 ) 0 0 0 0 9 - 0
80
H.A. Braun et al. / Neurocomputing 26}27 (1999) 79}86
In noise free, stationary, physical systems, periods as high as 95 [8] are experimentally detectable. The relatively recent advent of statistically and topologically based search algorithms [11}16,18] have made it possible to observe UPOs with statistical certainty in various biological preparations including electroreceptors [1] cold receptors [2] synaptic currents [7], bursting hippocampal neurons [10,15,17], human electroencephelographic records [9], and human cardiac dynamics [11]. But because biological preparations are typically non-stationary and include relatively large intensities of dynamical noise, only the "rst few periods have been experimentally observed in the biological preparations. Since higher-order periods are in general less likely to occur, we here focus only on the period-one orbits. In the following sections, we describe the "rst observations of UPOs in preparations from three radically di!erent types of temperature sensitive neurons: cat"sh electroreceptors, rat facial cold receptors and hypothalamic neurons from the rat brain paraventricular nucleus. We brie#y describe the data analysis method, techniques for preparing the neurons, experimental protocols and the results. We conclude with a brief summary.
2. Data analysis method All our data are in the form of a series of time intervals T, between neural action potentials. We seek particular sequences of six time intervals interspersed among the rest of the "le. These sequences are the signatures of an encounter of the general trajectory with a period-1 UPO. Beginning the analysis, we plot the data on a "rst return map, ¹ versus ¹ as shown in Fig. 1A. On this plot, a periodic point is one L> L which lies on the 453 line, that is ¹ ,¹ . An encounter with an unstable periodic L> L point is marked by a sequence of points which approach the 453 line with decreasing distances along a direction called the stable manifold followed immediately by a sequence which departs along the unstable manifold with increasing distances. An encounter is de"ned as the occurrence in the data "le of three points which approach
Fig. 1. Cat"sh electroreceptor. (A) Scatter plot of a complete record of time intervals. (B) An example encounter showing sequential points along the stable direction (squares, inward pointing arrows) and unstable direction (circles, outward arrows).
H.A. Braun et al. / Neurocomputing 26}27 (1999) 79}86
81
Fig. 2. Data from the cat"sh electroreceptor showing (top-to-bottom) scatter plots, histogram of encounters, intervals with K-values and "le segments indicated by the bar lengths, "ring rate and temperature time course. Stable and unstable sequences are indicated by squares and circles, respectively, with the mid-point shown by the large diamonds.
followed by three which depart. An example is shown in Fig. 1B. Searching the data "le results in N such occurrences. The points encompassing all identi"ed sequences are plotted over the complete "les in the examples shown in the top panel of Fig. 2. The noise results in the scattering of these collections of points, but in some cases (where
82
H.A. Braun et al. / Neurocomputing 26}27 (1999) 79}86
UPOs are clearly present in signi"cant concentrations) two populations, lying generally along the unstable and stable directions, can easily be identi"ed. The problem is the noise. If the dynamics contains noise, then the de"nition of an encounter can be ful"lled by accident. Therefore the question becomes: Is N a statistically signi"cant number? This question can be answered by comparing N with the average number 1N 2 found using the same encounter de"nition in randomized, or surrogate "les [19]. Proper surrogates will destroy the essential information contained in the de"nition of an encounter, that is, the sequential order of the seven time intervals. Thus all such sequences except random occurrences will be destroyed by randomly scrambling the order of the time intervals in the data "le. To obtain the mean, we scramble each data "le 100 times, "nd 1N 2 and the standard deviation p. We then examine the statistic, N!1N 2 K" p
(1)
in units of the standard deviation, sigma. K53 sigmas signi"es the detection of UPOs in the data "le with greater than 99% con"dence. The value of K indicates their concentration. Often statistically signi"cant determinations can be made rapidly from "les of only a few hundred time intervals, thus indicating that this algorithm is suitable for on-line determinations of the dynamical state of a preparation including transient changes as described below. Such algorithmic simplicity, and hence speed, is especially important in applications involving chaos control [6,15,3].
3. Experimental methods In all cases, a!erent action potentials were recorded extracellularly. The data analysis was performed o! line on the time intervals between the fast rises of sequential pairs of action potentials. For the peripheral neurons } electroreceptors and cold receptors } the surface temperature overlying the receptive "eld of the neuron was regulated and could be adjusted on demand either in discontinuous steps or swept over a range from 5 to 353C. In the case of the hypothalamus, the temperature of the #uid in a perfusion chamber containing the brain slice preparation could be adjusted in the same manner. The electroreceptor experiments were performed on live anesthetized cat"sh, Ictaluris nebulosus immobilized in a Lucite holder and arti"cially respired [1]. A small chamber containing thermally regulated water was sealed by O-ring over the receptive "eld. Recording was from the ampullary canal located in the dorsal head region using tungsten microelectrodes and standard extracellular electronics. The cold receptor experiments were performed on anaesthetized rats [2]. Recordings were made from single a!erent "bers of the infraorbital nerve with platinum electrodes. The temperature of the receptive "eld was controlled by a brass cylinder (3 mm diam. tip) over the range described above.
H.A. Braun et al. / Neurocomputing 26}27 (1999) 79}86
83
The hypothalamic recordings were made from brain slice preparations obtained from Sprague-Dawley rats. After decapitation, brain removal and conditioning, a block containing the hypothalamus was cut from the brain. Slices (400 lm thick, 5}7 mm diam.) were obtained and conditioned in oxygenated aCSF solution for at least one hour. The slice preparations were then mounted in a temperature controlled perfusion chamber. Single-unit recordings were made from the paraventricular nuclei (PVN) using glass micropipettes positioned with micromanipulators. File segments containing several hundred to three thousand time intervals were analyzed for the presence of period-one UPOs. As the following data demonstrate, the analysis method described here is suitable for examining the transient appearance and disappearance of UPOs in response to step and/or periodic variations in the temperature of as little as $13C.
4. Results Fig. 2 shows some example results from experiments on the cat"sh electroreceptor, though a more complete account has been made elsewhere [1]. The top panel shows four "rst return maps. The complete data "les from which these maps were constructed are shown by the open diamonds. The selected encounters are shown by enlarged symbols: squares indicating the stable directions and circles the unstable direction. The middle panel shows the temporal record of all time intervals ID (small diamonds), together with the time interval sequences comprising the identi"ed encounters in enlarged symbols (squares, stable; circles, unstable directions; large diamonds, the interval pair closest to the "xed point). The solid bars above show the "le segments analyzed and the respective K-values. The bottom panel shows the temperature time course. Note the response to the step temperature increase (30}353C) at about 85 s. Prior to this change, the value K"1.6 indicates no statistically signi"cant concentration of UPOs. However, the larger K-values after the change as well as the histograms of identi"ed encounters E, plotted above indicate the appearance of large concentrations of UPOs in response to the temperature change. These large concentrations (K"9.9 and 10.2), however, accommodate after about 175 s to insigni"cant values. The return maps in the top panel echo these "ndings: only the center two for the largest K-values show two well-de"ned populations of encounter points. In this case, the appearance of UPOs is clearly a transient phenomenon. We note that the mean "ring rate F, shown in a lower panel, indicates very little sensitivity to the temperature step. Some results of our experiments with the rat facial cold receptor [2] are shown in Fig. 3. Here, we illustrate the response of the neuron to a sinusoidal temperature stimulus of only $13C at an average temperature of 293C. The temporal record of all time intervals ID, is shown with bars and K-values above. The identi"ed encounters are again indicated by enlarged symbols. The histogram E, in the top panel shows that the encounter events indicating large concentrations of UPOs are
84
H.A. Braun et al. / Neurocomputing 26}27 (1999) 79}86
Fig. 3. Rat facial cold receptor showing (top-to-bottom) histogram of encounters, intervals with K-values and bars delineating the analyzed "le lengths, "ring rate and temperature time course.
phase locked to the stimulus sine wave albeit with a phase shift of about 3p/4, that is, the large concentrations appear on the downward phase of the temperature cycle. In this case the "ring rate F, shows a similar behavior though it seems to lag the UPO concentration somewhat. The UPOs enter during the increasing phases of the "ring rate cycle. Fig. 4 shows some results obtained from a hypothalamic neuron in the PVN. In this case, a linearly increasing temperature ramp is applied. Note the bifurcation to bursting beginning at approximately 200 s. In this case the K-values and histogram show that a large concentration of UPOs enters the dynamics at about 110 s after the ramp begins prior to the bursting bifurcation. After the bifurcation the neuron accommodates and the UPOs disappear. The "ring rate shows no indication of this behavior.
5. Summary In summary, we have demonstrated the existence of period-one UPOs in three diverse types of temperature sensitive neurons. In response to step, periodic or ramp temperature stimuli, the UPOs enter and depart transiently.
H.A. Braun et al. / Neurocomputing 26}27 (1999) 79}86
85
Fig. 4. Paraventricular nucleus neuron of rat brain with data arranged as in Fig. 3.
References [1] H.A. Braun, K. SchaK fer, K. Voigt, R. Peters, F. Bretschneider, X. Pei, L. Wilkens, F. Moss, Lowdimensional dynamics in sensory biology 1: thermally sensitive electroreceptors of the cat"sh, J. Comput. Neurosci. 4 (1997) 335}347. [2] H.A. Braun, M. Dewald, K. SchaK fer, K. Voigt, X. Pei, F. Moss, Low-dimensional dynamics in sensory biology 2: facial cold receptors of the rat, J. Comput. Neurosci., in press. [3] D.J. Christini, J.J. Collins, Controlling nonchaotic neuronal noise using chaos control techniques, Phys. Rev. Lett. 75 (1995) 2782}2785. [4] P. Cvitanovic, Invariant measurement of strange sets in terms of cycles, Phys. Rev. Lett. 61 (1988) 2729}2732. [5] P. Cvitanovic, Periodic orbits as the skeleton of classical and quantum chaos, Physica D 51 (1991) 138}152. [6] W.L. Ditto, S.N. Rauseo, M.L. Spano, Experimental control of chaos, Phys. Rev. Lett. 65 (1990) 3211}3214. [7] P. Faure, H. Korn, A nonrandom dynamic component in the synaptic noise of a central neuron, Proc. Natl. Acad. Sci. USA 94 (1997) 6506}6511. [8] E.R. Hunt, Stabilizing high-period orbits in a chaotic system-the diode resonator, Phys. Rev. Lett. 67 (1991) 1953}1955. [9] M. Le Van Quyen, J. Martinerie, C. Adam, F.J. Varela, Unstable periodic orbits in human epileptic activity, Phys. Rev. E. 56 (1997) 3401}3411.
86
H.A. Braun et al. / Neurocomputing 26}27 (1999) 79}86
[10] L. Menendez de la Prida, N. Stollenwerk, J.V. Sanchez-Andres, Bursting as a source for predictability in biological neural networks activity, Physica D 110 (1997) 323}331. [11] K. Narayanan, R.B. Govindan, M.S. Gopinathan, Unstable periodic orbits in human cardiac rhythms, Phys. Rev. E 57 (1998) 4594}4736. [12] X. Pei, F. Moss, Characterization of low dimensional dynamics in the cray"sh caudal photoreceptor, Nature 379 (1996a) 618}621. [13] X. Pei, F. Moss, Detecting low dimensional dynamics in biological experiments, Int. J. Neural Systems 7 (1996b) 429-435. [14] D. Pierson, F. Moss, Detecting periodic unstable points in noisy chaotic and limit cycle attractors with applications to biology, Phys. Rev. Lett. 75 (1995) 2124}2127. [15] S.J. Schi!, K. Jerger, D. Duong, T. Chang, M.L. Spano, W.L. Ditto, Controlling chaos in the brain, Nature 370 (1994) 615}620. [16] P. So, E. Ott, S.J. Schi!, D.T. Kaplan, T. Sauer, C. Grebogi, Detecting unstable periodic orbits in chaotic experimental data, Phys. Rev. Lett. 76 (1996) 4705}4708. [17] P. So, E. Ott, T. Sauer, B.J. Gluckman, C. Grebogi, S.J. Schi!, Extracting unstable periodic orbits from chaotic time series data, Phys. Rev. E 55 (1997) 5398}5417. [18] P. So, J.T. Francis, T.I. Neto!, B.J. Gluckman, S.J. Schi!, Periodic orbits: a new language for neuronal dynamics, Biophys. J. 74 (1998) 2776}2785. [19] J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, J.D. Farmer, Testing for nonlinearity in time series: the method of surrogate data, Physica D 58 (1992) 77}94.
