Biol. Cybern. 86, 1±14 (2002)

Mechanisms for lateral turns in lamprey in response to descending unilateral commands: a modeling study Alexander K. Kozlov1;2 , Fredrik UlleÂn2 , Patriq Fagerstedt2 , Erik Aurell1;3 , Anders Lansner1 , Sten Grillner2 1

SANS/NADA, KTH ± Royal Institute of Technology, LinstedtsvaÈgen 5, 100 44 Stockholm, Sweden Department of Neuroscience, Karolinska Institute, 171 77 Stockholm, Sweden 3 Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden 2

Abstract. Straight locomotion in the lamprey is, at the segmental level, characterized by alternating bursts of motor activity with equal duration and spike frequency on the left and the right sides of the body. Lateral turns are characterized by three main changes in this pattern: (1) in the turn cycle, the spike frequency, burst duration, and burst proportion (burst duration/cycle duration) increase on the turning side; (2) the cycle duration increases in both the turn cycle and the succeeding cycle; and (3) in the cycle succeeding the turn cycle, the burst duration increases on the non-turning side (rebound). We investigated mechanisms for the generation of turns in single-segment models of the lamprey locomotor spinal network. Activation of crossing inhibitory neurons proved a sucient mechanism to explain all three changes in the locomotor rhythm during a ®ctive turn. Increased activation of these cells inhibits the activity of the opposite side during the prolonged burst of the turn cycle, and slows down the locomotor rhythm. Secondly, this activation of the crossing inhibitory neurons is accompanied by an increased calcium in¯ux into the cells. This gives a suppressed activity on the turning side and a contralateral rebound after the turn, through activation of calcium-dependent potassium channels.

1 Introduction Locomotion allows an animal to purposefully move around in its environment. The fundamental rhythmic locomotor movements are generated by spinal neuronal circuits, while supraspinal centers adapt the movements to the needs and goals of the animal by supplying adequate descending commands for postural adjustments, alterations in speed, and steering (Grillner 1985; Orlovsky et al. 1999). Such modi®cations are accomCorrespondence to: A. K. Kozlov (Tel.: +46-8-7907784, Fax: +46-8-7900930 e-mail: [email protected])

plished by modifying the activity of either the locomotor musculature itself or other muscle groups. The generation of the basic locomotor pattern has been studied in several invertebrate and vertebrate in vitro models. The spinal central pattern generator for swimming movements in the lamprey is well characterized in terms of identi®ed interneurons, their synaptic connectivity, and post-synaptic e€ects (Rovainen 1983; Grillner et al. 1995). Electrical stimulation of the brainstem or chemical activation of the spinal cord in vitro by L-glutamate agonists evoke alternating ventral root bursting activity (®ctive locomotion) in the same frequency range as the muscular activation in the intact animal during swimming (Cohen and WalleÂn 1980; Poon 1980; McClellan and Grillner 1984; Brodin et al. 1985). The locomotor central pattern generator consists of two halves containing pools of excitatory interneurons (EINs), which drive motoneurons (MNs), and noncrossing inhibitory interneurons (e.g., lateral interneurons (LINs), see next paragraph). The two sides of the network are coupled to each other by both local and caudally projecting reciprocal inhibitory neurons, here referred to as commissural inhibitory neurons (CINs). Di€erent mechanisms have been proposed to terminate the burst activity on one side. Burst termination can be achieved by a delayed activation of ipsilateral inhibitory mechanisms. In certain early models this inhibition was provided by LINs (Rovainen 1983), which have long caudally projecting axons that do inhibit CINs, thus tentatively providing an important burst terminating function. However, this function can also be carried out by cellular adaptation mechanisms induced by Ca2‡ entry into CINs and EINs, as explored by Hellgren et al. (1992), who tested the network with and without LINs. During turns in the horizontal plane (lateral turns), the muscular activity on the side towards which the animal is turning (the turning side) increases, producing a larger force against the water on that side, which results in a turn (McClellan and Hagevik 1997). Apart from an increased burst amplitude on the turning side, there are also changes in the locomotor rhythm during

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lateral turns, both in adult (Fagerstedt and UlleÂn 2001) and in larval lamprey (McClellan and Hagevik 1997). These turn-related changes have been examined both in vivo and in vitro in a brain-spinal cord preparation with skin and muscle tissue attached to the head (McClellan and Grillner 1984; Fagerstedt and UlleÂn 2001). Asymmetric ventral root burst activity with the same changes both in timing and intensity of the bursts as during an in vivo turn (a ®ctive turn) was evoked during ®ctive locomotion by electrical stimulation of the skin on one side of the head (McClellan and Hagevik 1997; Fagerstedt and UlleÂn 2001; Fagerstedt et al. 2001). By de®nition, the term cycle will be used to denote the time period between the onset of one burst and the onset of the subsequent burst on the same side. The cycle on the turning side that includes the start of the turning command will be referred to as the turn cycle. In summary, both real and ®ctive turns are characterized by the following alterations of the locomotor rhythm during a turn (data not shown, see Fig. 1 as a particular example): 1. An increase of the MN spike frequency (ventral root burst intensity) on the turning side. 2. An increased duty cycle duration. 3. An increase of the burst duration, leading to an increase also in burst proportion (burst duration/cycle duration) on the turning side. 4. Rebound; i.e., an increase of the burst duration and burst proportion of that burst on the side opposite to the turning side, which occurs immediately after the increased burst in the turn cycle. 5. A decrease of the interburst interval, which is the time interval between bursts on the two sides, both in the turn cycle and in the following cycle. The main goal of the present work is to investigate the possible mechanisms whereby the unilateral excitatory turn command could accomplish these changes in the locomotor rhythm. For this purpose we use a number of models of the lamprey spinal locomotor pattern generator that have been considered in the earlier literature (see Sect. 2 and the Appendix for further details). For purposes of illustration, turn commands lasting several cycles were used in the simulations of the present study. However, the same e€ects could in all cases be evoked in a single cycle by using shorter commands, as in the in vitro situation (data not shown). 2 Methods Several models of the spinal locomotor generating circuits have been considered in the literature. In level of complexity they range from simple abstract oscillator models (McClellan and Hagevik 1997) to detailed models of neuronal populations where the aim is to make all cellular and populational parameters as realistic as possible (Ekeberg et al. 1991; Hammarlund and Ekeberg 1996).

Fig. 1A,B. Fictive turn in the in vitro preparation. A Electrical stimulation to one side of the head of a brain±spinal cord preparation of the lamprey (L and R skin electrodes) evoked asymmetric burst activity in the ventral roots (L rostral VR15 , L caudal VR30 , R rostral VR15 and R caudal VR30 ) during ®ctive locomotion induced by 1.5 mM D-glutamate in the spinal cord compartment (D-glutamate pool). B The skin stimulus (Stim) elicited an increased burst duration and intensity in the ventral roots contralateral to the stimulus (Contra VR15 and Contra VR30 ), followed by a rebound consisting of increased burst intensity and duration in the ipsilateral ventral roots (Ipsi VR15 and Ipsi VR30 ). The cycle duration during the response is also increased, but to a lesser extent than the increase in burst durations, leading to increased burst proportions on the contralateral side during the turn, followed by increased burst proportions on the ipsilateral side in the succeeding cycle. Rostral ventral roots are at the 15th segment and caudal roots are at the 30th segment of the spinal cord

The present study used models at an intermediate level of complexity implementing two di€erent hypotheses for the generation of the locomotor pattern that are currently discussed in the literature. One is dependent on a speci®c cell type, the LIN for burst termination (WalleÂn et al. 1992). The other implements slow calcium dynamics provided by N -methyl-D-aspartate (NMDA) channels (Brodin et al. 1991). Both models share the following set of characteristics: 1. Each segment consists of two identical and reciprocally connected (via CINs) hemisegments (see Fig. 2B).

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The latter two cell types were not included in the spiking model (Fig. 2B). 3. All neurons of a particular type in a segment of the spinal cord are considered to constitute a single functional unit. The activity of a unit is interpreted as a proxy for the total activity of the pool of individual neurons of the corresponding type. 4. In the spiking model, each unit is modeled by equations similar to a ®ve-compartment model of a single neuron, but with adjusted parameter values re¯ecting the higher aggregated frequency of spikes from the pool of individual neurons as a whole. A more detailed description of the models, with surveys of the directly relevant literature, is given in Sect. 2.1. For precise parameter values, see the Appendix.

2.1 Non-spiking network model dependent on LIN burst termination

Fig. 2A±C. Characteristics of the two models used. A Network used in the non-spiking model. The network included four units in each hemisegment: an MN unit, M, acting as a pure output element; an EIN unit, E, exciting the other units on the same side; a CIN unit, C, inhibiting units in the contralateral hemisegment; and an LIN unit, L, acting as a burst terminator by inhibiting units in the same hemisegment (see text). B Network used in the spiking cell model. Here only two units were used in each hemisegment: an EIN unit, E, and a CIN unit, C. For details, see text and the Appendix. C Swimming frequency in the spiking model at di€erent levels of AMPA drive, with and without the NMDA drive being present (parameters ka and kn , respectively, see the Appendix; ka ˆ 3±6, kn ˆ 0:3, or kn ˆ 0:0 in arbitrary units, a:u:). Using an AMPA drive in this range, the spiking network was capable of generating locomotion at 1±8 Hz

2. The following neuronal types are included in the model: (a) EINs, which excite all other cell types on the same side. (b) CINs, which inhibit all other cell types on the opposite side. (c) LINs, which inhibit CINs on the same side. (d) MNs, which are excited by EINs and act as pure output elements, and do not participate in pattern generation.

Here, a simpli®ed non-spiking neuron model was used, in which the model neuron acts as a leaky integrator with a saturating synaptic transfer function (Fig. 2A). All synaptic inputs were subject to a dendritic time delay. A spike-frequency adaptation mechanism with a di€erent time constant was also included. Parameters were tuned to the input±output relations and spike frequency of biological neurons (Ekeberg et al. 1991; WalleÂn et al. 1992). It should be noted, though, that NMDA properties were not included in this model, which meant that the model lacked slow components in its dynamics provided by slow mixed Na‡ =Ca2‡ currents activated by NMDA receptors. For a detailed description of the neuron model, see Ekeberg (1993); for a recent review, see Ekeberg and Grillner (1999); for a related model with modulation of the slow afterhyperpolarization, see Lansner et al. (1998) and UllstroÈm et al. (1998). The simulated locomotor neuronal network included all four units discussed above, that is the proxies for the EIN, CIN, LIN, and MN cell pools. The EIN unit excited all units on the same side of the spinal cord; the CIN unit inhibited units on the contralateral side, and the LINs strongly inhibited units on the same side, thus ± in this model ± acting as powerful terminators of the locomotor bursts on the same side. The MN units acted as pure output elements and did not participate in the generation of the locomotor rhythm. The activity level of an MN unit was interpreted as a measure of the burst intensity (spike frequency) of the corresponding hemisegment (see Sect. 3). For a more detailed description of synaptic strengths and the rostrocaudal extent of connections, see Ekeberg (1993). Locomotion was maintained by a general excitatory drive to all units. Turn commands were simulated as additional unilateral or bilateral excitatory or inhibitory biases. These could be given either to all units in a hemisegment or to one or several speci®c units (see Sect. 3). Commands of di€erent magnitude and duration were investigated.

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2.2 Spiking cell model Each functional unit was modeled using a single-cell model, with membrane properties and ionic channels as in Ekeberg et al. (1991). These were equipped with six types of ionic channels: K‡ ; Na‡ , NMDA, low-voltageactivated Ca2‡ (LVA), high-voltage-activated (fast) 2‡ ‡ Ca2‡ (Ca2‡ …K‡ AP ), and a Ca -regulated K Ca †. There 2‡ were two Ca pools, one linked to the (fast) Ca2‡ AP channel and one linked to the (slow) NMDA channel. LVA channels were implemented as in TegneÂr et al. (1997), and NMDA channels as in Brodin et al. (1991). The compartmentalization followed WalleÂn et al. (1992); that is, the initial segment was separated from the soma, and the active membrane properties responsible for spike generation were distributed between these two compartments. All computer simulations of the model were perfomed with GENESIS software (Bower and Beeman 1998). Synaptic conductances were modeled as conductance increases for Cl and cations, and as saturating at high spike frequencies (TraÊveÂn et al. 1993). Inhibitory synapses were fast while excitatory synapses had both fast a-amino-3-hydroxy-5-methyl-4-isoxazolepopionic acid ((AMPA)/kainate) and somewhat slower voltage-dependent NMDA components, as in Brodin et al. (1991) and Ekeberg et al. (1991). The connectivity of the single segment was as in WalleÂn et al. (1992), but simpli®ed in that LIN and MN units were not included in the model, which thus consisted of four functional units, proxies for one EIN, and one CIN cell pool in each hemisegment (Fig. 2B). The parameters relating to membrane properties and synaptic weights were as in Hellgren et al. (1992) and related speci®cation ®les (J. Hellgren, personal communication; see the Appendix). Each EIN unit excited the ipsilateral CIN unit, while each CIN unit inhibited the contralateral EIN and CIN units. The EIN unit was regarded as the output element of a hemisegment, since the EIN cell pool normally drives the MN pool (see Sect. 3). Excitatory in¯uences to spinal locomotor neurons from the brainstem are mediated by both fast pure Na‡ currents through channels that are gated by AMPA receptors, and slow mixed Na‡ =Ca2‡ currents activated by NMDA receptors. In our model we do not include controlling neurons for the initiation of locomotion and turning in the brainstem explicitly, but mimic their output by excitatory drives acting through synaptic couplings on the neural elements in a segment. Inclusion of slow NMDA receptor properties is one of the main di€erences between this and the non-spiking model discussed in Sect. 2.1. We investigated the importance of this by evoking turns in the model with two di€erent excitatory drives: a mixed AMPA±NMDA drive and a pure AMPA drive. The model could produce alternating burst frequencies over a large range in both conditions (Fig. 2C). For parameter values, see Sect. 3 and ®gure legends. In di€erent simulations, three types of manipulation were used in attempts to evoke ®ctive turns:

1. Turn commands were simulated by injecting DC currents or as changes of the excitatory drive (cf. above) to one or both types of units in a hemisegment. Commands of di€erent magnitude and duration were investigated. 2. The strength of the crossing inhibitory connections from the CIN unit on the turning side were transiently increased, to give an asymmetric reciprocal inhibition during a turn. 3. The neuronal circuitry was complemented with an additional unit similar to the CIN unit …CINturn †, to asymmetrically increase the crossing inhibition during a turn (see Sects. 3 and 4). 3 Results 3.1 Non-spiking network model 3.1.1 Changes in burst intensity. An excitatory command to all units in one half of the spinal cord increased the MN burst intensity on the stimulated side, while the MN burst intensity on the contralateral side remained constant (Fig. 3A). An increase in burst intensity could also be evoked by a command directed to the MN unit alone (Fig. 3B), in which case the cycle duration and burst duration remained una€ected, since the MN unit provides no input to the rhythm generator (Grillner et al. 1995). A command selectively in¯uencing only the EIN unit or only the CIN unit did not increase the ipsilateral MN burst intensity (data not shown). An extra activation of the proxy for the MN cell pool thus seems to be necessary and sucient for an increase in burst intensity in this model. It should be noted that all observed changes in MN burst intensity ± evoked by the turn command ± were much smaller than in vivo (see Figs. 1 and 3A). An excitatory command of the same magnitude as the general excitatory bias maintaining locomotion caused an increase in MN burst intensity of only around 15%. The increase in MN burst intensity as a function of the turn-command magnitude followed a roughly linear slope: to obtain a 50% increase in MN burst intensity, a turn command more than three times larger than the general excitatory bias was required (data not shown). 3.1.2 Changes in the timing of the locomotor rhythm. A unilateral excitatory command addressed to all units decreased the cycle duration (Fig. 3A). When the command was of the same magnitude as the general excitatory bias, the decrease was around 30% (data not shown). Similar non-turn-like increases in swimming speed were evoked by any command that excited the burst-terminating LIN unit. Since the LIN unit exerts a powerful burst-terminating e€ect in this model (see Sect. 4), the only way to avoid a decrease in cycle duration during a turn proved to be to exclude the LIN unit from the turn command. Even so, the increase in cycle duration remained small (less than 2% ; data not shown).

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excitatory descending command to the EIN and CIN units with either a strong decrease of the general excitatory bias or an additional inhibitory command to the LIN unit, increases in cycle duration of 20% or more could be evoked. The burst duration also increased slightly under these conditions, but this increase was considerably smaller than the increase in cycle duration, and the net e€ect was thus a non-turn-like decrease in burst proportion. No modi®cations of this unilateral command, by adding speci®c excitatory or inhibitory e€ects in di€erent combinations, resulted in turn-like changes in all parameters of the locomotor rhythm. To achieve a realistic turn the EIN and CIN units of the non-turning side had to be depressed even further. Increasing the excitation of the existing CIN on the activated side did not provide sucient crossing inhibition, suggesting activation of additional inhibitory e€ects on the other side during turns. The added crossing inhibition was simulated as increased inhibitory bias to the EIN and CIN units, and resulted in a turn with increased burst duration, cycle duration, burst proportion, and burst intensity (Fig. 3C). However, the changes were much smaller than in vivo. Two other properties of ®ctive turns proved not to be reproducible in this model: rebound, and the decrease in the interburst interval. The rebound requires cell properties with long time constants, and these are not present in this model. It was also not possible to study the latter phenomenon, since the model does not display a realistic interburst interval during normal locomotion. We conclude that this model of the spinal locomotor generator, which heavily relies on the LIN unit to terminate bursts, needs to be supplemented with both additional cell properties and a modi®ed connectivity in order to produce all the key aspects of lateral turns. The model showed, however, that the increase in MN burst intensity during turns is possible to obtain by a direct descending activation of the MN cell pool. Fig. 3A±C. Simulated turns in the non-spiking model. The output of the left and right MN units in the ®rst segment is shown. The application of the turn command is indicated with a horizontal bar. Turn commands were applied as excitatory or inhibitory biases added onto the general excitatory bias driving the locomotion. For details, see text. A Excitatory turn command to all units on one side of the spinal cord. On the stimulated side, an increased burst intensity as well as a decreased cycle duration is seen. B Excitatory turn command, a€ecting only the MN unit on one side. An increase in burst intensity without any e€ects on the timing of the locomotor pattern is seen. C Bilateral turn command, combining an excitation of the EIN, CIN, and MN units and inhibition of the LIN unit on the turning side, with opposite e€ects on cell units on the non-turning side. In this way, turnlike changes in burst intensity, cycle duration, and burst proportion could be evoked, albeit still with smaller amplitude than in vivo

To increase the cycle duration further, selective inhibition of the LIN unit had to be introduced. This corresponds to either evoking a selective inhibitory descending command (Wannier et al. 1995) or to modifying the functional connectivity of the model by adding segmental neurons which inhibit the LIN cell pool (Buchanan and Grillner 1988). By combining the

3.2 Spiking cell model 3.2.1 Changes in burst intensity in the standard network. In the spiking model, burst termination does not depend on the activity of a speci®c cell type (LIN; see Sect. 3.1.2). The activity on one side is terminated by the intrinsic membrane properties of the CIN units causing them to stop inhibiting the other side. This results in a locomotor pattern with non-zero interburst intervals and an overall more natural burst proportion (Cohen and WalleÂn 1980; Poon 1980). The functional units also include cellular processes with long time constants (i.e., NMDA and calcium properties; see Sect. 2 and the Appendix) which, as we will see, makes this model more suitable for simulations of ®ctive turns (see Sect. 3.2.3). Since this model does not contain a proxy for the MN cell pool, the motor output of the central pattern generator is approximated by the activity of the EIN unit, since the corresponding cell pool drives the MN pool directly (see Sect. 2).

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The contribution of the central pattern generator to the MN burst intensity was hence estimated from the spike frequency of the EIN unit. If a unilateral turn command acted only on the EIN unit (Fig. 4A,B), or on the EIN and CIN units simultaneously (Fig. 4A,D), burst intensity increased by a moderate amount (Fig. 5A). A command exclusively directed to the CIN unit had practically no e€ect on burst intensity (Fig. 4A, C). In this respect, the behavior of the spiking model was quite similar to that of the non-spiking model, where a turn command directed to the output elements was necessary to get an increase in burst intensity. The mathematical model used for the EIN and CIN units (see the Appendix) operates close to the upper limit of obtainable spike frequency, which can therefore not easily be further increased (see Sect. 4). 3.2.2 Changes in the timing of the locomotor rhythm in the standard network. Turn commands to the CIN

Fig. 4A±D. Burst pattern during turn commands with di€erent target speci®city in the spiking model. A 3.2 AMPA/0.3 NMDA drive was used. Turn commands were simulated by current injections of 0.15 nA. The outputs of the left and right EIN units are shown. A Normal burst pattern without any turn command. B Unilateral turn command to the EIN unit. A moderate increase in burst intensity (spike frequency) and a decrease in the cycle duration are seen. C Unilateral turn command to the CIN unit. The burst intensity remained practically una€ected, while the cycle period and the burst proportion on the stimulated side increased. D Unilateral turn command to both the EIN and the CIN units. An increase in burst intensity, cycle duration, and burst duration on the stimulated side is seen

functional unit alone, or to the EIN and CIN units simultaneously, produced an increase in cycle duration (Figs. 4, 5B), while commands acting only on the EIN unit produced a shortened cycle duration (Figs. 4, 5B). In addition, commands to the CIN unit, or to the EIN and CIN units simultaneously, produced a large increase in burst duration on the stimulated side, and a decrease on the opposite side (Figs. 4, 5C,D). In contrast, turn commands acting on the EIN unit only had small e€ects on the burst duration (Figs. 4, 5C,D). The relative e€ects on burst duration were larger than the corresponding e€ects on cycle duration, leading to an increased burst proportion on the turning side (Fig. 5E) and a decreased burst proportion on the non-turning side (Fig. 5F). Interburst intervals decrease, and can even get negative; thus both sides may ± for some short time ± burst simultaneously, in agreement with experimental observations (see Fig. 1). All these results ± increased burst duration, cycle duration and burst proportion, and decreased interburst intervals ± are in clear agreement with the experimental observations (see Sect. 1 and Fagerstedt and UlleÂn 2001). However, the e€ect of stimulation in the model is still generally weaker than that observed in vivo and is only observed over a limited dynamic range; e.g., sometimes accompanied by a response with the opposite sign to small input currents (cf. Fig. 5B). 3.2.3 Importance of NMDA properties for turn generation in the spiking model. In the spiking model, however, the rebound was observed with all the di€erent turn commands in a mixed NMDA ± AMPA drive (0.3 NMDA/3.2 AMPA) of the network. In order to test the dependence on di€erent cellular properties, the excitatory drive to the locomotor network was changed to a pure AMPA drive (0 NMDA/3.2 AMPA). It was concluded that the rebound was dependent on the presence of NMDA properties, since no rebound could be observed in the pure AMPA drive (Fig. 6A,C). The voltage-dependent NMDA channels are permeable to Ca2‡ ions and open when the cells are depolarized in the presence of glutamate in the active phase. The level of intracellular Ca2‡ in the CIN unit is increased above normal during the turn and remains elevated for a longer time during the silent phase (Fig. 6B). Ca2‡ activates hyperpolarizing K‡ Ca currents and the CIN unit is then depressed for a longer time than normal, prolonging the silent phase after the turn. If the CIN unit lacks NMDA properties, the silent phase after the turn is not prolonged, and the rebound is absent (Fig. 6D). 3.2.4 Modi®cations of the connectivity in the standard network. In the standard model, turn commands to the CIN unit produced all the turn-like changes except for an increase of the burst intensity. This could, on the other hand, be achieved by adding excitation to a unit acting as a proxy for the MN pool. The e€ects of stimulation were weaker than in vivo, and some parameters changed in the opposite direction at low stimulus strengths (i.e., cycle duration; Fig. 5C). It was

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Fig. 5A±F. Turn amplitude as a function of turn command amplitude in the spiking model. All turn parameters are calculated from the activity of the EIN unit. Values of di€erent parameters of the locomotor pattern are shown for commands of di€erent intensity, directed to the EIN unit, the CIN unit, or both units together. A Ipsilateral burst intensity (spike frequency). A moderate increase of burst intensity with increasing stimulation strength is seen for commands given to the EIN unit, or to both the EIN and the CIN units. B Cycle duration. Commands to the CIN unit, or to both the EIN and the CIN units, increased the cycle duration, although sometimes a small decrease could be seen in response to small input currents (see text). For current strengths above 0.23 nA (commands to the CIN unit only), or 0.18 nA (commands to both the CIN and EIN

units), the turn command evoked continuous tonic activity on the stimulated side, and cycle duration, burst duration and burst proportion were thus determined by the duration of the turn command. C, D Ipsilateral and contralateral burst duration. Commands to the CIN unit, or to both the EIN and the CIN units, increased the ipsilateral burst duration and decreased it on the contralateral side. Commands to the EIN unit had little e€ect on the burst duration. E, F Ipsilateral and contralateral burst proportion. The increase in ipsilateral burst duration by commands directed to the CIN unit, or to both the EIN unit and the CIN unit (see C, D, and text), was larger than the accompanying increase in cycle duration, giving an increased ipsilateral burst proportion. The contralateral burst proportion decreased

only possible to obtain turn-like changes over a limited dynamic range, perhaps due to a smaller frequency range of the functional units above the normal spiking frequency during ®ctive locomotion. This hypothesis was tested in two ways: In the ®rst set of simulations, we increased the strength of the e€erent synapses from the CIN unit on the turning side

transiently during the turn command. A turn command directed to the EIN and CIN units, or to the CIN unit only, produced more pronounced turns (data not shown). In the second approach, we introduced an additional functional unit CINturn with similar connectivity and synaptic weights as the CIN unit (Fig. 7A). The CINturn did not participate in the generation of the

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Fig. 7A±E. E€ects of changing the connectivity of the spiking model. A Modi®ed network with one extra CIN unit, CINturn (CT ). The CINturn unit received no input from the other units and was silent during normal locomotion. It had inhibitory connections to the contralateral cell units in the same way as the CIN unit. B Normal ®ctive locomotion without any turn command. C Simulated turn with a unilateral excitatory command (stimulation strength 0.1 nA) to the EIN and the CIN units in 3.2 AMPA/0.3 NMDA drive. D Simulated turn with a unilateral excitatory command to the EIN, the CIN, and the CINturn units (same drive as in C). The stimulation strength to the CINturn unit was approximately ten times higher (1.5 nA) than to the other cell units, to compensate for the lack of excitatory input from the EIN unit. The turn amplitude is considerably higher than in C. E Cycle duration during the turn as a function of turn command amplitude for commands given to the CIN and the EIN units only, or to the CIN, the EIN, and the CINturn units. In the latter case, the e€ects on cycle duration were larger and the turn cycle duration increased monotonically as a function of command amplitude (see text)

duration and burst proportion than those produced by a turn command to the EIN and CIN units only (Fig. 7B± D). The increase in cycle duration was monotonic when CINturn was activated, in contrast to the situation in the standard network. 4 Discussion

Fig. 6A±D. The dependence of the contralateral rebound on NMDA properties. Turn commands were applied to both the EIN and the CIN units, with a current strength of 1.5 nA. A Simulated turn in 3.2 AMPA/0.3 NMDA drive. The activity of the ipsilateral and the contralateral EIN units is shown. A clear rebound, i.e., prolonged burst duration on the contralateral side after the turn, is seen. B Ca2‡ dynamics of the CIN unit during the turn in A. Note the higher-thannormal Ca2‡ concentration at the end of the turn, which gives a larger activation of K‡ Ca channels and thus a longer inhibition of the turn side during the rebound (for further details, see text). C Simulated turn in 3.2 AMPA drive only (no NMDA). No rebound is seen. D Calcium dynamics of the turn in C. Without NMDA drive, the calcium concentration at the end of the turn burst is not higher than normal

normal locomotor rhythm, i.e., did not receive synaptic in¯uences from the ipsilateral EIN unit, and was activated only by the turn command. This additional unit need not necessarily be interpreted as representing a distinct population (see Sect. 4). A turn command directed to the EIN, CIN, and CINturn units simultaneously produced an larger cycle

In this work we investigate turning in single-segment models of the spinal network in lamprey. We de®ne turns to include speci®c characteristics that have been recently established experimentally, listed in the Sect. 1, of which the most dicult to simulate have been an increased cycle duration of the basic swimming pattern and a rebound, both in response to descending unilateral excitatory commands. With a less restrictive de®nition of what constitutes a turn, e.g., increased burst intensity as measured in the MN cell pool, or actual turns in a combined neuromechanical model (Ekeberg 1993), models which fail the criteria used in this paper may nevertheless be acceptable qualitatively, as recently reviewed in Ekeberg and Grillner (1999). We have used two broad classes of models, both of which are built on functional units acting as proxies for cell pools. In one class, the units are leaky integrators, i.e., objects with an output of a real amplitude, which is intended to model a burst from a population of functionally similar neurons. Furthermore, in this class the activity in one hemisegment is terminated by a speci®c inhibitory unit, representing the LIN cell pool. In the other class, the units are objects with an output of spikes. Both of these models have been amply described in the literature (Ekeberg et al. 1991; WalleÂn et al. 1992; Ekeberg and Grillner 1999), and have previously been validated in normal swimming mode, in particular to support oscillations at di€erent frequencies. One reason behind this work is to show how experimental results on turns can be used as a falsi®cation tool to eliminate some proposed models. A second reason is to deduce plausible mechanisms of turn control, and to

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Fig. A1. Five-compartment model of a cell in the locomotor network

circumstances. More precisely, the non-spiking model can reproduce the observed behavior of increased output amplitude at the MN pool, which is what determines muscle activity and turn-like motion of the animal as a whole, but not the characteristics of the EIN or CIN pools controlling the MN pool. The spiking cell model was ®rst modi®ed in terms of on which functional unit a command acts (EIN, CIN, or both EIN and CIN). Commands acting only on the EIN unit failed to reproduce the experimental observations of increased cycle duration in any version of the model. On the other hand, commands acting only on the CIN unit failed to show increased burst intensity. A case-by-case inventory, detailed in Sect. 3, concludes that no version of the spiking model with the modi®cations listed above matches all of the experimental observations, but that the best are commands acting on both the EIN and CIN units. 4.2 Role of cellular properties

propose possible experimental tests of these mechanisms. 4.1 Discrimination between models We ®rst turn to the model-elimination part, and summarize our results as follows: the non-spiking model does not show turns in any variant tried, while the spiking model does so under a well-de®ned set of

A main di€erence between the spiking model and the non-spiking model was that the latter could not reproduce the rebound phenomenon after the end of the stimulus, suggesting that membrane properties with long time constants are necessary components of models of ®ctive turns. In the spiking model, the units contain ion channels, such as the NMDA channels that are permeable to Ca2‡ ions, the intracellular level of which is regulated by ion pumps with slow time constants. Ca2‡ ions activate Ca2‡ -dependent potassium channels …K‡ Ca † which hyperpolarize the neuron through an e‚ux of K‡ ions. Variants of the nonspiking model with slow adaptation have recently been presented (Lansner et al. 1998), in the context of mechanisms for locomotor generation in the lamprey spinal cord (UllstroÈm et al. 1998). It would be of interest in future work to test if these variants of the non-spiking model can reproduce rebound and other characteristics of turns. In the spiking model, NMDA channels in CINs are voltage dependent and either synaptically activated by glutamate from the EINs during the active phase in each swim cycle, or activated by the bath-applied NMDA. During a turn, the CINs receive additional excitation from the descending turn command which

10 Table A1. Ionic channels used in the model. Two currents, ICaAP and ICaNMDA , are considered not to contribute to the total membrane current, and are included for calculation of the calcium concentration only Channel type, ionic current

Erev

Fast sodium current INa ˆ  gNa m3 h…V ENa †

Conductance dependencies

0.050

Potassium current IK ˆ  gK n4 …V EK †

)0.080

Low-voltage-activated calcium current ILVA ˆ  gLVA m3 h…V ECa †

0.150

Bath-activated AMPA current bath ˆ gAMPA …V EAMPA † IAMPA Bath-activated NMDA current bath INMDA ˆ gN p…V ENMDA †

0.000

Fast calcium current ICaAP ˆ  gCF q5 …V ECa †

0.150

Slow calcium current gCS p…V ECS † ICaNMDA ˆ  Calcium-dependent potassium current IK…Ca† ˆ  gK…Ca† ‰CaŠ…V EK †

0.020

0.000

)0.080

Coe€. am bm ah bh Coe€. an bn Coe€. am bm ah bh

Eq. (A4) (A4) (A4) (A3) Eq. (A4) (A4) Eq. (A4) (A4) (A4) (A3)

A

B )0.001 0.02 0.001 )0.002 B )0.0008 0.0004 B )0.0045 0.0045 0.0078 )0.0048

V0 )0.045 )0.054 )0.045 )0.041 V0 )0.045 )0.035 V0 )0.058 )0.061 )0.063 )0.061

Coe€. ap bp Coe€. aq bq bath p is same as in INMDA

Eq. (A2) (A2) Eq. (A4) (A4)

A 700 10.08 A 0:08  106 1000

B

V0 0.000 0.000 V0 0.015 0.015

Conc. ‰CaAP Š ‰CaNMDA Š ‰CaŠ ˆ 5‰CaAP Š ‡ ‰CaNMDA Š Ca currents: ICaAP , ICaNMDA  gCS ˆ 1 gCF ˆ 

Eq. (A5) (A5)

BCa =S 670  109 1:107  109  gN

sCa 1/20 1/2.6

0:2  106 0:06  106 0:08  106 400 A 0:02  106 5000 A 0:02  106 0:05  106 100 30

0.017 )0.017 B )0.011 0.0005

syn Table A2. Postsynaptic currents. Currents contributing to the membrane potential (A1) are listed in the ®rst column; current ICaNMDA is provided for calculation of calcium concentration ‰Casyn Š only NMDA

Synapse type, contributed currents

Esyn

Conductance dependencies

Inhibitory synapse syn Iinh ˆ gsyn Einh †s…t† inh …V

)0.085

q 1/0.0001

r 1/0.010

Tdur 0.0005

sdel 0.0014

Excitatory AMPA/kainate synapse syn ˆ gsyn Ekain †s…t† Ikain kain …V

0.000

q 1/0.0001

r 1/0.010

Tdur 0.0005

sdel 0.0033

Excitatory NMDA synapse syn INMDA ˆ gsyn ENMDA †s…t† N p…V

0.000

q 1=0:005

r 1=0:100

Tdur 0:020

sdel 0:0033

Coe€. Eq. ap (A2) bp (A2)

A 700 10.08

B 0.017 )0.017

V0 0.000 0.000

Conc. ‰Casyn NMDA Š syn Ca current: ICaNMDA ˆ gCS p…V ECS †s…t† bath p is same as in INMDA , gCS ˆ 1

Eq. (A5)

BCa 1:107
109  gsyn N

sCa 1/2.6

syn syn IK…Ca†NMDA ˆ gsyn KN ‰CaNMDA Š…V

EK †

)0.080

opens more NMDA channels, leading to a larger Ca2‡ in¯ux. The Ca2‡ levels in CINs on the turning side increase above normal during the turn, and remain elevated longer during the silent phase, which through hyperpolarization by K‡ Ca channels will keep the CINs depressed for a longer time. The CIN unit on the op-

posite side, being more inhibited than usual by the crossing inhibitory activity from the turning side, is depleted of intracellular Ca2‡ to a larger extent than normal. This will inactivate more K‡ Ca channels and increase the excitability the CIN unit on the nonturning side after a turn, leading to a rebound. This

11 Table A3. Parameters of cell compartments Compartment

Area relation

Passive properties

Ionic channels

sm

ga

Current

Conduct.

Initial segment

0:1Ssoma

0.012

300

INa IK bath IAMPA

gNa ˆ 835 gK ˆ 415 gAMPA ˆ 1
ka

Soma

Ssoma

0.006

4.2

INa IK ILVA IK…Ca† bath IAMPA INbath

gNa ˆ 50:1 gK ˆ 83 gLVA ˆ 25  gK…Ca† ˆ 40
10 gAMPA ˆ 1
ka gN ˆ 75
kn

bath IAMPA

gAMPA ˆ 1
ka

Proximal dendrite

4Ssoma

Medial dendrite

4Ssoma

Distal dendrite

4Ssoma

0.012

0.012

0.012

4.2

4.2

4.2

bath IAMPA

bath IAMPA

Table A4. Units of parameters and variables used in the model. The dimension of BCa corresponds to dimensionless calcium concentration (1/coulomb) Parameter

Unit

Description

S Cm sm g g I V E m; h; n; p; q [Ca] ka ; kn

m2 F s S Sm A V V 1 1 1

BCa sCa q r D T

C 1 s s 1 s 1 s s

array membrane capacitance membrane time constant conductance speci®c conductance current membrane potential reversal potential gate variables (dimensionless) calcium concentration (dimensionless) bath activation levels for AMPA and NMDA (dimensionless) calcium in¯ux rate calcium decay rate synapse activation time constant synapse decay time constant synaptic delay duration of transmitter release

2

mechanism is here suggested to be essential for the rebound. This prediction would be experimentally testable by blocking K‡ Ca channels pharmacologically during in vitro turns. 4.3 Connectivity of the network In a second set of modi®cations we increased the contralateral inhibition during a turn, either by asymmetrically changing the synaptic weights of the mutual CIN cross-inhibition, or by including an additional functional unit, CINturn , which was only

Synapses

gAMPA ˆ 1
ka

9

exc. NMDA  gsyn N ˆ 2:6  10  gsyn N ˆ 2:6  10  gK…Ca† gsyn KN ˆ 

inh.  gsyn inh ˆ 364  10  gsyn inh ˆ 208  10  gsyn inh ˆ 364  10  gsyn inh ˆ 208  10 exc. AMPA  gsyn A ˆ 32:5  10  gsyn A ˆ 26:0  10

9

(E ! C) (E ! E)

9

9 9 9 9

9 9

(C ! C) (C ! E) (CT ! C) (CT ! E) (E ! C) (E ! E)

gAMPA ˆ 1
ka

activated during turns. This modi®cation proved to be a robust mechanism to e€ect turns with regard to increased cycle duration; i.e., in contrast to above. In addition, we also performed numerical experiments using direct negative current injection in the contralateral CIN unit (data not shown), which displayed very similar behavior. This last modi®cation uses a descending command which is neither excitatory nor unilateral, but it provides further proof that increased inhibition of the contralateral CIN cell pool, by whatever mechanism, is a simple way of reproducing key experimental observations. Whether this is the actual mechanism used could be tested by recording from crossing interneurons during ®ctive turns, such as in Fagerstedt et al. (1998). We now turn to an interpretation of these results. The functional units in the model that we have used (the EIN and CIN) are described by equations originally proposed for single neurons, but with modi®ed parameter values such that the spiking frequency from the unit matches that of the total spike frequency of the corresponding cell population (for details, see the Appendix). This construction has some side e€ects. The most relevant for the present investigation is that these proxies for entire cell populations behave as large cells bursting at or close to their saturation level. Their spiking frequency can therefore not be increased easily. This explains why we observed, at best, a comparatively weak increase in burst intensity; this feature is practically built into these models. Preliminary data (Kozlov et al. 2000) from simulations where the EIN and CIN units in this investigation were replaced by 50 cells, each ®ring at a 50-fold-lower spiking frequency during locomotion, shows that with a sucient dynamic range for the response to a descending command, a turn can be achieved without the addition of units.

12

4.4 Conclusions In this paper we have investigated mechanisms for the control of ®ctive turns (i.e., turn-like behavior in preparations of the isolated nervous system) in lamprey. In vivo turns, where sensory feedback from, for example, spinal stretch receptors is likely to also play a role, may be governed by further mechanisms of control. The e€ects of such mechanisms are, however, not known in comparative detail at present. We found strengthened crossed inhibition to be an ecient means of inducing an increased cycle duration, burst proportion, and burst duration ± all three key experimental characteristics of in vitro turns. This most likely corresponds to an increased activity of crossing inhibitory neurons, since a facilitation of the inhibitory synapses from these neurons would take place on the opposite side. This increased activity may be achieved by an increased aggregate spike frequency or recruitment, either of the crossing inhibitory neurons involved in the generation of the locomotor pattern (i.e., the CIN unit) or of such neurons that are not involved in the generation of the locomotor pattern but are speci®cally activated during turns (i.e., the CINturn unit). Here and in Kozlov et al. (2000), both of these mechanisms have been veri®ed to be e€ective. In fact, the main conclusion of this study is that inhibition of the contralateral side, by whatever mechanism, reproduces the experimental characteristics of in vitro turns. Acknowledgements. This work was supported by the Swedish Institute (A.K.), the Swedish Natural Science Research Council (MA 1778-333; E.A.), the Swedish Medical Research Council (MFR 3026 and 11544; P.F and S.G.), the Swedish Research Council for Engineering Sciences (F.U.), and the Swedish Brain Foundation (F.U.).

We used a ®ve-compartment cell model, consisting of an initial segment, a soma, and a three-compartment dendritic tree, as shown on Fig. A1. The initial segment, which contains about 90% of the fast sodium and potassium channels, was responsible for spike generation. The remaining active ionic channels were located on the soma. The dendrites were used mainly for synaptic couplings. Tonic bath activation was applied to all compartments. A.1 Compartment model

Ia…j†

…A1†

Eleak † V …j† †

ˆ ga …V

where gm is a membrane leak conductance, ga is an axial conductance, Eleak is the leak reversal potential, and V …j† is the membrane potential of neighboring compartment j. The active properties in a description of a compartment are the ionic channels activated by the membrane potential, the internal calcium concentration, the synapses, and the chemicals in the bath. A.2 Ionic currents Currents through active ionic channels are described by equations of the Hodgkin±Huxley type: I ˆ G…V

Erev † ˆ gmk1 hk2 …V

Erev †

where Erev is the reversal potential, G ˆ gmk1 hk2 is the channel conductance, m and h are the gate variables (taking values between 0 and 1) which determine closed and open states of the channel, and k1 and k2 are integers determined by the cooperativity of the gating switch. The dynamics of a voltage-dependent gate variable x is modeled by a ®rst-order kinetic equation x†

bx …V †x

where ax …V † and bx …V † are rate coecients. The voltage dependence of a rate coecient y…V † (ax …V † and bx …V † in the equation above) is approximated by one of the following forms   V V0 …A2† y…V † ˆ A exp B   V y…V † ˆ A= exp

y…V † ˆ A…V

The dynamics of the membrane potential of a compartment was determined by the equation X X X dV ‡ Ileak ‡ Iion ‡ Isyn ‡ Ia…j† ˆ Iinj dt syn j ion

Ileak ˆ gm …V

dx ˆ ax …V †…1 dt

Appendix: Detailed description of the spiking model at the cellular level

Cm

where V is the membrane potential, Cm is the membrane capacitance, Ileak is the leak current, Iion is the current through ionic channel, Isyn is the postsynaptic current, …j† Ia is the axial current from a neighboring compartment j, and Iinj is the injected current. The passive properties of a compartment are its area, capacitance, and two currents ± the leakage and the axial current:

V0 B



 ‡1

  V V0 †= exp

…A3† V0

B



 1

where A; B; and V0 are constants. A calcium-dependent gate variable x ˆ f …‰CaŠ†

…A4†

13

follows calcium concentration ‰CaŠ in an internal pool:

A.6 Units

d‰CaŠ ˆ BCa ICa dt

All values in the model are given in SI units; see Table A4.

‰CaŠ sCa

…A5†

where ICa is the calcium current, and BCa and sCa are constants. For parameter values of the ionic channels, see Table A1. A.3 Postsynaptic currents Currents activated by synapses are also modeled by equations of the Hodgkin±Huxley type: Isyn ˆ g…t†…V

Esyn †

where Esyn is the reversal potential, g…t† ˆ gsyn pk s…t† is the channel conductance, s…t† is a synapse activation variable, and p is the gate variable modeling a Mg2‡ block if necessary (k ˆ 1 or k ˆ 0). Synaptic activation is modeled by the leaky integrator equation  ds q…1 s†; 0 < t …tspike ‡ sdel † < Tdur ˆ …A6† rs; otherwise dt where q and r are growth and decay time constants, Tdur is the duration of transmitter release, sdel is the synaptic delay, and tspike is the time instant of a spike event in the presynaptic membrane. For parameter values of synapses, see Table A2. A.4 Compartments The electrical properties of a compartment vary linearly with the compartment area S (see below). The constant proportionality coecients are the speci®c capacitances CM ˆ Cm =S and the speci®c conductances g ˆ g=S. The value of CM is 0:01. The membrane time constant sm is RM CM , where RM ˆ 1=gm is the speci®c membrane resistance, and thus gm ˆ CM =sm . The equilibrium potential of a passive leakage current through the membrane is Eleak ˆ 0:070 V. For parameter values of the passive membrane properties and conductances of ionic channels, see Table A3. A.5 Cells Neurons of both types (CIN and EIN) are described by similar ®ve-compartment models which di€er only in the cell sizes and synaptic connections. The area of a spherical cell soma is S ˆ pd 2 , where the cell diameter d is 30  10 6 m for CIN cells and 20  10 6 m for EIN cells. AMPA bath activation (parameter ka in Table A3) of CIN cells which are silent during locomotion and can become active during turns (CINturn ) is reduced by a factor 0:014.

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