Neurocomputing 44–46 (2002) 381 – 389
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A neural ensemble model of the respiratory central pattern generator: properties of the minimal model Witali L. Dunin-Barkowski ∗ , Andrew T. Lovering, John M. Orem Department of Physiology, Texas Tech University Health Sciences Center, 3601 4th Street, STOP 6551, Lubbock, TX 79430-6551, TX, USA
Abstract A computational model of the respiratory central pattern generator is considered in this work. The model is based on mutual inhibition of two core neuron pools, which neurons accommodate as the result of Ca in2ux and activation of Ca-dependent K + channels (J. Neurophysiol. 77 (1997) 1994). The model yields known respiratory neuron types and explains by ensemble features the observable properties of these respiratory neurons. Original procedures for the adec 2002 Elsevier Science B.V. All rights quate presentation of the modeling data are described. reserved. Keywords: Respiratory central pattern generator; Neural ensemble; Respiratory neurons
1. Introduction Modeling of neural mechanisms of the respiratory control is one of the oldest topics in computational neuroscience ([3,4,7], and many others). Nevertheless, the problem is still far from being solved [2]. Even the fundamental issues, such as whether the respiratory rhythm arises from a network or intrinsic pacemaker neurons remain obscure [15]. Interpretation of any experimental data on the activity of respiratory neurons depends on the neural mechanisms of the respiratory central pattern generator (RCPG). A RCPG model is needed to understand experimental data. This fact and the absence of a well accepted RCPG model were leitmotifs of the present attempt. On the basis ∗
Corresponding author. Tel.: +1-806-743-2522; fax: +1-806-743-1512. E-mail address:
[email protected] (W.L. Dunin-Barkowski).
c 2002 Elsevier Science B.V. All rights reserved. 0925-2312/02/$ - see front matter PII: S 0 9 2 5 - 2 3 1 2 ( 0 2 ) 0 0 3 8 5 - 5
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of this attempt we have formulated a number of problems and principles which we believe to be important in the Enal model.
2. Methods The model consists of a number of neurons and neuronal Ebers. The model of a neuron consists of: (a) mechanism for summation of incoming synaptic inputs. We have so far used just two types of synapses: inhibitory and excitatory. The model is mostly standard [7]. The only original element in our model is that we use an economical way of introduction in the network of diversity of synaptic delays. Conductivity changes, imposed by a presynaptic impulse, are presented by a pair of exponents of diGerent signs and the same amplitude. Time constants for these exponents are the same for all synapses on a given neuron, but they are diGerent for neurons; (b) there are only three types of electromotive forces in the neurons: inhibitory with an equilibrium potential of −90 mV; excitatory with an equilibrium potential of −12 mV and leaking with an equilibrium potential of −70 mV; (c) a neuron becomes excited when its membrane potential reaches a threshold. At this moment the excitation-related potassium conductivity receives an increment. At the same moment intracellular calcium concentration increases. Both decay permanently with diGerent time constants—small (10 ms) for excitation-related potassium conductivity and large (500 ms) for calcium concentration. Calcium concentration immediately governs calcium-dependent potassium conductivity [16]. We use two types of neurons: those with and those without calcium-dependent potassium conductivity. The :bers are variables, that are at each time moment set to be either excited or not. The excitation is provided by a random mechanism, that provides an independent random choice for each Eber at each time moment with probabilities of excitation set at diGerent levels for diGerent Eber groups. The connections from neurons and Ebers to neurons are provided with a connections matrix. Its elements are chosen randomly once when the network is initialized and they are held constant. Connection probabilities are speciEed below. The algorithm of network operation is straightforward: at each time step for each neuron its incoming connections-vector is multiplied by a vector of state of neurons and Ebers. Then its membrane potential is updated, subject to all membrane current, described above. We use the exponential integration method for time-variable variables [7]. The time-step value in our model is 1 ms. The core of the RCPG model consists of the two neuron pools that represent decrementing inspiratory and decrementing expiratory neurons. These two neuron groups are mutually inhibitory. We do not introduce connections inside the groups. The remaining neurons in the model receive inhibitory inputs from the core neuron pools in diGerent
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Table 1 Neuron parameters
Neuron types\ Eequ Parameters
mem
PI
PI
Ca
Ca
Ee
1e
2e
Ei
1i
2i
(mV) (ms) (a.u.) (ms) (a.u.) (ms) (mV) (ms) (ms) (mV) (ms) (ms)
Core pool −70 Non-core pool −70
10 10
1 1
10 10
0.05 0
500 —
−12 −12
15 15
3 3
−90 −90
15 15
3 3
Here: Eequ —neuron’s equilibrium potential, mem —neuron’s time constant, PI —change of membrane conductivity after an impulse, PI —time constant of after-impulse conductivities, Ca —an increase of calcium concentration after an impulse, Ca —time constant of calcium concentration changes, Ee —equilibrium potential of excitatory synapses, 1e and 2e —average values of time constants of positive and negative exponents for excitatory synapse, Ei , 1i , and 2i same as previous for inhibitory synapse. Table 2 Parameters of interconnection matrix between neurons and between Ebers and neurons
Neuron pool type (elements no.)
Idec (136)
Eaug (136)
Bi-phasic (68)
Edec (204)
Iaug (204)
Idec (136)
======
inhib Prb = 0:5 w = 0:13
inhib Prb = 0:15 w = 0:13
inhib Prb = 0:2 w = 0:13
inhib Prb = 0:15 w = 0:13
inhib Prb = 0:2 w = 0:13
inhib Prb = 0:1 w = 0:156
inhib Prb = 0:15 w = 0:13
======
inhib Prb = 0:3 w = 0:13
======
excite Prb = 0:2 w = 0:0022
======
excite Prb = 0:2 w = 0:0022
======
======
======
======
======
======
======
excite Prb = 0:2 w = 0:0022
======
excite Prb = 0:2 w = 0:0022
excite Prb = 0:2 w = 0:0022
excite Prb = 0:2 w = 0:0022
excite Prb = 0:2 w = 0:0022
excite Prb = 0:2 w = 0:0022
excite Prb = 0:2 w = 0:0022
Edec (204) Fibers 1 (204) Fibers 2 (204) Fibers 3 (204) Fibers 4 (204)
excite Prb = 0:2 w = 0:0022
Here w is a synaptic weight; for Ebers excitation probability at each time moment was 0.45.
proportions. Neurons of all groups receive tonic excitation from input Ebers. Tables 1 and 2 summarize the model parameters. 3. Results Fig. 1 demonstrates the activity patterns of the system. Each horizontal line here represents activity of one neuron; impulses are denoted by dots. The order in which
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Fig. 1. Raster of activity of RCPG neurons. Each horizontal line of dots corresponds to one neuron. Each dot represents neuron’s Ering. Horizontal and vertical solid lines—1 s and 100 neurons.
neurons are sampled is arbitrarily randomized. It is evident here that the network indeed oscillates. There are alternating longer and shorter periods of neuronal activity. The shorter is natural to relate to inspiration, while the longer, to expiration. Simultaneous transitions of masses of neurons from one respiratory phase (inspiration) to another is notable in this Egure. There are more or less salient changes of global states of the system inside each of the respiratory phases. The moments of the phase switch are so well deEned in Fig. 1 that they suggest abrupt changes in the number of active neurons, connected with the phase switch. Fig. 2 demonstrates that this is not the case. Here the time course of two integral parameters of the network activity is presented. The Erst is the number of active neurons in a current time moment. Its time course, although irregular, is relatively smooth: there
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Fig. 2. Time course of integral characteristics of RCPG activity. Top trace—number of neurons, active in a current time moment; bottom trace—the smallest order number of the neuron (see Fig. 3 for deEnition of the order number) that is active in a current time moment. Horizontal solid line at the bottom—2 s.
are no dramatic changes of the number of excited neurons at the moment of phase switching. The abrupt changes are obvious, however, in the plot of the smallest order number (SON) of the neuron that is excited at the current time moment (SON-plot, bottom curve of Fig. 2). This is not surprising because the neurons in the network are enumerated consecutively, one neuron group after the other (i.e. inspiratory, expiratory and bi-phasic). The SON time course also reveals a presence of sub-phases in the network. It can be seen that at the Erst one-third of the inspiratory period and in the Erst one-forth of the expiratory period, de2ections of the SON curve from the current average value are substantially less, than in the remaining part of the corresponding respiratory phase. This can be seen also in Fig. 1, that shows more silent neurons in the Erst third of the inspiratory phase and the Erst quarter of the expiratory phase. The network in fact seems to have not two general phases, inspiration and expiration, but four phases: early inspiration, late inspiration, early expiration and late expiration.
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Fig. 3. Types of neuron enumeration. Dashed straight line—original enumeration (used in Fig. 2); light gray—randomized enumeration (used in Fig. 1); solid line—original enumeration enhanced with the ordering according to the order parameter (used in Fig. 4). Abscissa—the original order; ordinate—the permuted order.
However, with an appropriate enumeration (Fig. 3) we can see that it is not the case (Fig. 4). The patterns of enumeration of neurons which have been used for displaying the results in Figs. 1, 2 and 4 are presented in Fig. 3. Here are: (a) the original generic enumeration (bottom trace in Fig. 2); (b) the randomized enumeration (Fig. 1); and (c) the original enumeration, enhanced with neuron reordering according to the value of an integral of the neuron activity over the initial period of modeling (Fig. 4). The display in Fig. 4 is the most informative for evaluation of the model performance. This display demonstrates distinct neuron groups that more or less Et well all the known types of respiratory neurons (Idec , Iaug , Edec , Eaug , E–I , bi-phasic [1,6]), except for inspiratory throughout and expiratory throughout neurons. The moments of transition from inspiration to expiration and vice versa are much more evident in Fig. 4, than in Fig. 1. Note also that many Iaug and Eaug neurons discharged impulses at the beginning of inspiration and expiration, respectively. Fig. 5 demonstrates patterns of the intracellular potentials that are observed in the model. It is evident that the neural potential traces are physiologically plausible. Tracks of decrementing neurons (both expiratory and inspiratory) are similar. These two types are the only motors of the system. The other neurons are passive followers of the ‘instructions’ from these core neuron pools. The expiratory augmenting neurons produce an impulse at the transition to expiration and after a delay increase their Ering frequency. The only example on this display of the augmenting inspiratory neuron
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Fig. 4. Ordered raster of activity of RCPG neurons. Notations and the data are the same, as at the Fig. 1. Neurons are displayed according to their neuron pool type and according to the value of the order parameter inside the neuron pools.
demonstrates another activity pattern. It starts Ering before the transition to inspiration and gradually increases its Ering rate. The E–I neurons have two active phases—one in the expiratory phase, the other in the inspiratory. One of these neurons Eres exactly at the moments of respiratory phase switching. The silent neurons show sharp waves of depolarization at moments of phase switching. When other parameters are Exed and the number of neurons is increased (proportionally in all groups), the oscillation frequency of the RCPG decreases (not illustrated). Frequency also decreases when the background non-rhythmic synaptic in2ow to all neuron types increases (not illustrated). Also at other constant parameters, the duration of a respiratory phase increases with an increase in number of neurons in the generating core neuron pool of that phase (not illustrated).
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Fig. 5. Patterns of neuron’s intracellular potentials in the RCPG model. Types of neurons from top to bottom—expiratory decrementing (2 tracks); expiratory augmenting (2 tracks); E–I (2 tracks); inspiratory decrementing (2 tracks); silent neurons (two tracks); E–I (1 track); inspiratory augmenting (1 track). Time marks at 1 s intervals.
4. Discussion As demonstrated above, the minimal ensemble-based model of the RCPG produces almost all types of respiratory neurons in the network. The two absent neural types (inspiratory and expiratory throughout) are easy to obtain by introducing neurons that are inhibited by both types (decrementing and augmenting) of neurons of the opposite phase. Of course, the same result could be obtained with excitation from both major types of neurons of the same respiratory phase (e.g., augmenting and decrementing inspiratory neurons). However, in the minimal model of the RCPG we prefer to use inhibition for connections between the generic types of neurons, leaving excitation for external (‘tonic’) in2uences—just to have only one source (inhibition) of all neuron Ering properties. We have demonstrated also the importance of the proper display of activity of the modeled neurons. Fig. 1 presents the activity of the system that is not sorted. We have demonstrated, that in the model there exists an order parameter that makes neural activity easily observable (Fig. 4). It should be emphasized that the system behavior is determined by interaction of two pools of mutually inhibitory neurons. No post-inhibitory rebound properties were considered. Nevertheless, many augmenting neurons showed single post-inhibitory impulses after a delay. That means that the ‘rebound’ Ering and augmenting pattern of neuron discharge do not require speciEc intrinsic membrane mechanisms (cf. [16]). The time scale of the oscillations is determined by the Ca concentration dynamics. All excitatory in2uences to the system are tonic. Under these conditions the total number of active neurons in the system varies relatively smoothly. Nevertheless, there is a sharp depolarization wave at the moments of respiratory phase switchings [14]. This depo-
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larization wave is evidently the property of the ensemble-based system. There is no necessity to consider special switching neurons that become active at phase transitions [14]. We have established that respiratory system properties can be dependent on ensemble instead of single neurons properties. One of the most important properties of the ensemble is its size, that is the number of neurons in each core neuron pool. Unfortunately, there is not even an approximate estimation of the size of CRPG in nature. Our simulations, however, demonstrate, that the chosen numbers (about 200 neurons in each core neural pool) are suPcient to provide many properties of the system. We consider our simulations as a necessary step in obtaining an instrument for analysis and interpretation of our data on the activity of single medullary respiratory neurons in awake and sleeping cats [5,8–13]. In these applications we need to analyze system behavior over large segments of time (tens of minutes), when the system generates thousands of breaths as compared to about ten, readily available with the present simulation system. Evidently this problem requires a super-computer-based approach for its solution. References [1] U.J. Balis, K.F. Morris, J. Koleski, B.G. Lindsey, Simulations of a ventrolateral medullary neural network for respiratory rhythmogenesis inferred from spike train cross-correlations, Biol. Cybern. 70 (1994) 311–327. [2] R.J. Butera, J. Rinzel, J.C. Smith, Models of respiratory rhythm generation in the pre-Botzinger Complex, J. Neurophysiol. 80 (1999) 382–415. [3] A.N. Chetaev, Neural Networks and Markov Chains, Nauka Publishers, Moscow, 1985, p. 125. [4] W.L. Dunin-Barkowski, Activity oscillations in a simple closed neural network, BioEzika 15 (1970) 374–378. [5] W.L. Dunin-Barkowski, J.M. Orem, Suppression of diaphragmatic activity during spontaneous ponto-geniculo-occipital waves in cat, Sleep 21 (7) (1998) 671–675. [6] A. Gottschalk, M.D. Ogilvie, D.W. Richter, A.I. Pack, Computational aspects of the respiratory pattern generator, Neural Comput. 6 (1993) 56–68. [7] R.J. MacGregor, Neural and Brain Modeling, Academic Press, San Diego, 1987, p. 643. [8] A. Netick, J.M. Orem, W. Dement, Neuronal activity speciEc to REM sleep and its relation to breathing, Brain Res. 120 (1977) 197–209. [9] J.M. Orem, Neuronal mechanisms of respiration in REM sleep, Sleep 3 (3=4) (1980) 251–267. [10] J.M. Orem, Excitatory drive to the respiratory system in REM sleep, Sleep 19 (10) (1996) S154–S156. [11] J.M. Orem, Augmenting expiratory neuronal activity in sleep and wakefulness and in relation to duration of expiration, J. Appl. Physiol. 85 (4) (1998) 1260–1266. [12] J.M. Orem, R.H. Trotter, Behavioral control of breathing, News Physiol. Sci. 9 (1994) 228–232. [13] J.M. Orem, A.T. Lovering, W.L. Dunin-Barkowski, E.H. Vidruk, Endogenous excitatory drive to the respiratory system in rapid eye movement sleep, J. Physiol. 527 (2) (2000) 365–376. [14] D.W. Richter, Neural regulation of respiration: rhythmogenesis and aGerent control, in: R. Gregor, U. Windhorst (Eds.), Comprehensive Human Physiology, Vol. II, Springer, Berlin, 1996, pp. 2079–2095. [15] D.W. Richter, K.M. Spyer, Studying rhythmogenesis of breathing: comparison of in vivo and in vitro models, Trends Neurosci. 24 (8) (2001) 464–472. [16] I.A. Rybak, J.F.R. Paton, J.S. Schwaber, Modeling neural mechanisms for genesis of respiratory rhythm and patterns, J. Neurophysiol. 77 (1997) 1994–2039.
