- Email: [email protected]
Role of frequency-dependent weighing of inputs on frequency regulation of a pacemaker-driven rhythm
ARTICLE IN PRESS
Neurocomputing 69 (2006) 1103–1107 www.elsevier.com/locate/neucom
Role of frequency-dependent weighing of inputs on frequency regulation of a pacemaker-driven rhythm Inbar Saraf-Sinik, Yair Manor Life Sciences Department, Ben-Gurion University, Beer Sheva 84105, Israel Available online 3 February 2006
Abstract We examined the frequency-dependent efficiency of positive and negative inputs onto pacemaker neurons of the pyloric rhythm. When neuromodulation to the pyloric circuit was blocked, pyloric neurons became quiescent. In the pacemaker neurons, we injected series of sinusoidal currents of different frequencies. At slow frequencies, voltage peak amplitudes were significantly larger than troughs. This trend reversed at higher frequencies. A computational model suggests that rapid A- and slower T-type currents can produce this excitability profile, and that a frequency dependence preference of negative and positive inputs impinging onto pacemaker neurons can play an important role in homeostatic stabilization of the rhythm. r 2006 Elsevier B.V. All rights reserved. Keywords: Pacemaker; Frequency-dependent excitability; Oscillation; Stomatogastric ganglion; Pyloric rhythm
1. Introduction Many rhythmic activities are mediated by central pattern generators (CPGs) [3]. A key question is what properties endow a CPG with regularity and flexibility at the same time. It is intuitive to accept that some types of external neuromodulation could control the CPG frequency by differentially strengthening or weakening inhibitory versus excitatory inputs impinging onto CPG neurons. One source of complication to consider is the frequency-dependent excitability of these neurons. In sub-threshold regimes, the voltage responses to positive or negative currents of the same amplitude are similar, hence a frequency-dependent transfer function (impedance) is a reliable tool to quantify the frequency-dependent excitability properties [2,7]. However, in biological neurons voltage responses to currents of opposite signs are not necessarily symmetrical, especially when these currents are large. Hence, a measure of frequency-dependent excitability that distinguishes between inhibitory and excitatory inputs may be more informative. Corresponding author.
E-mail address: [email protected] (Y. Manor). 0925-2312/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2005.12.054
The well-studied pyloric rhythm is an example of a CPG which pattern is affected by inhibitory and excitatory inputs [5]. When descending neuromodulation to the pyloric network is intact, this CPG generally produces a stable oscillation (1 Hz, hereafter referred to as fnatural). Occasionally, however, the rhythm frequency can vary between 0.3 and 2.5 Hz. The oscillations are generated by a group of electrically coupled conditional oscillators, the anterior burster (AB) neuron and the two pyloric dilator (PD) neurons. The oscillatory activity is propagated to follower neurons via chemical inhibitory and electrical synapses (see diagram in Fig. 2). Two followers, the lateral pyloric (LP) neuron and the ventricular dilator (VD) neuron, provide negative and positive inputs back to the pacemaker neurons. Within a range of cycle periods, these two neurons may regulate the pyloric frequency in complementary ways [9]. In this work we developed a measure of frequencydependent excitability that distinguishes between positive and negative inputs. Focusing on the PD neuron, we examine the possibility that frequency-dependent preference of negative and positive inputs impinging on pyloric pacemaker neurons contributes to the stabilization of the pyloric rhythm.
ARTICLE IN PRESS 1104
I. Saraf-Sinik, Y. Manor / Neurocomputing 69 (2006) 1103–1107
2. Methods
2.3. Modeling
2.1. Electrophysiology All experiments were done on the isolated stomatogastric nervous system of the lobster Homarus americanus. Intracellular recordings of PD neurons were done with sharp microelectrodes. PD neurons were isolated from their neighbors by blocking chemical synapses with bath application of picrotoxin (5 106 M), and killing electrically coupled cells with photoinactivation techniques [4]. Descending neuromodulation was impaired by blocking all electrical activity in the stomatogastric nerve (stn), usually by bath application of a solution containing tetrodotoxin (107 M). Following this procedure, the pyloric rhythm was abolished and all pyloric neurons became quiescent.
The network model consisted of three interconnected cells: an auto-rhythmic cell (representing the PD neuron) and two follower cells (representing the VD and LP neurons). The PD model was coupled to the VD model with a rectifying electrical synapse, and to the LP model via reciprocal inhibition. All three cells were implemented as conductance-based models, comprising leak, transient sodium, delayed rectifier, T-like calcium and A-like potassium currents. The PD cell also included a ‘‘proctolin current’’ (a ligand-activated voltage-dependent inward current), which represented the effect of neuromodulation [8]. Time-dependent variables were numerically integrated with the fourth-order Runge–Kutta method, as described in [1].
2.2. Construction of the frequency-dependent excitability profile
3. Results
In a quiescent PD neuron, we injected a ‘‘sweep’’ current waveform (amplitude: 73 nA). The sweep waveform consisted of a consecutive series of 13 sinusoidal waves of discrete frequencies (f 1 ¼ 0:1, f 2 ¼ 0:2; . . ., f 10 ¼ 1:0, f 11 ¼ 1:5; . . ., f13 ¼ 3.0 Hz). At any frequency fi, a sine wave was injected for 2 (io8) or 4 (iX8) cycles. To eliminate transient effects and allow a smooth transition between any two discrete frequencies fi and fi+1, following the 2 (or 4) cycles of fi a single transition cycle was injected. The frequency of this transition cycle was continuously changed from fi to fi+1. Fig. 1 shows a portion of the sweep waveform current. For each discrete frequency fi, on the voltage response (Fig. 1A) we measured the peak and trough amplitudes of the last cycle (open symbols). These values were then divided by current amplitude, and normalized with respect to input resistance. This procedure yielded the two functions — Z+(f) (for peaks) and — Z(f) (for troughs), that together defined the excitability profile of the cell.
Fig. 1. The sweep current waveform stimulus and voltage response of the PD neuron.
In the intact network, biological PD neurons showed a typical oscillatory activity (Fig. 2). Following synaptic isolation of the PD neuron, the rhythm in the PD neuron slowed down (from 1.0170.24 to 0.76 Hz70.02, n ¼ 2). We hereafter refer to the cycle frequency of the rhythm produced by the isolated PD neuron as fintrinsic ( ¼ 1/ Pintrinsic). When neuromodulation was blocked, oscillations disappeared. Fig. 2B reproduces these experimental observations in a network model. We tuned the parameters of the model such that, in the intact network, the three cells produced a 1 Hz rhythm. When feedback connections from the followers were disabled, the rhythm slowed down to 0.75 Hz. Bath application of tetrodotoxin, which blocked neuromodulation, was modeled by setting sodium and proctolin conductances in the PD model to 0. Under this condition, oscillations of network model stopped. The excitability profiles of 37 biological PD neurons were measured as described in Methods. Fig. 3A2 is an example of a PD voltage trace in response to intracellular injection of a current sweep waveform. The average excitability profile (Mean7SE, n ¼ 37) is shown in Fig. 3A1. Interestingly, within the stomatogastric ganglion we observed similar excitability profiles in the AB neuron only. In theory, the enhanced responsiveness at slow frequencies could be due to a slowly inactivating inward current (for example, a T-current), whereas a rapidly inactivating outward current (for example, an A-current) could explain the decreased responsiveness at high frequencies. In fact, in our experiments we observed that the characteristic frequency-dependence excitability profiles of biological PD neurons were disrupted in the presence of A- or Tcurrent blockers (data not shown). Hence, we reasoned that A- and T-current could be responsible for this characteristic excitability profile. Therefore, throughout this work the PD cell was modeled with a rapidly inactivating A-current (small th,A relative to Pintrinsic),
ARTICLE IN PRESS I. Saraf-Sinik, Y. Manor / Neurocomputing 69 (2006) 1103–1107
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Fig. 2. The pyloric rhythm: (A) biological and (B) computational model.
Fig. 3. The frequency-dependent excitability profile of a PD neuron: (A) biological and (B) model.
and a slowly inactivating T-current (large th,T relative to Pintrinsic). Fig. 3B2 shows the voltage response of an isolated PD model cell (simulated with a blocked neuromodulation, and with no feedback connections from follower neurons) to current injection of a sweep waveform. Here we extended the range of frequencies, starting from f1 ¼ 0.01 to f72 ¼ 50 Hz. Although frequencieso0.1 Hz and 43 Hz are not physiological, examination of the PD neuron responsiveness in this extended range of frequencies is informative, as detailed below. In the physiological range (o3 Hz), except for small discrepancies the excitability profile was similar to the one obtained in the biological PD neurons. There was essentially no effect of frequency on the troughs of PD responses (— Z): during the hyperpolarization phases the cell essentially acted as a low-pass filter with a cut-off frequency around 3 Hz. The effect of frequency on the peaks ( — Z+) can be explained by examining the effects
of frequency on the slow T-current and faster A-current. Let hA and hT represent the gating variables of A- and Tcurrent inactivation, respectively. At slow frequencies (‘‘a’’ in Fig. 3B1) depolarization is slow relative to th,A and th,T, and in both cases the currents inactivate (hA, hT-0) before they can activate. Hence, neither current is active. As f increases (‘‘b’’ in Fig. 3B1), depolarization rate is still slow relative to th,A, hence the A-current does not affect the peak (hA-0 during depolarization). However, depolarization rate is already comparable to th,T, and T-current activates before it inactivates (hT40 during depolarization). As a result, the T-current becomes larger and, as an inward current, it enhances the peak of the oscillation: — Z+ + increases. At f ¼ 0.3 Hz (‘‘c’’ in Fig. 3B1), — Z reaches a local maximum. A larger frequency means not only a faster depolarization, but also a shorter duration of the hyperpolarizing window: beyond ‘‘c’’, the hyperpolarizing window becomes short relative to th,T. The deinactivation of the T-current is incrementally smaller (hTo1 during
ARTICLE IN PRESS 1106
I. Saraf-Sinik, Y. Manor / Neurocomputing 69 (2006) 1103–1107
hyperpolarization), thereby moderating its effect on the peak, and — Z+ decreases. This trend continues until the Tcurrent can no longer deinactivate (hT0 during hyperpolarization): beyond this frequency, the T-current does not contribute to the peak. This occurs around f ¼ 0.7 Hz (‘‘d’’ in Fig. 3B1), where — Z+ intersects with — Z. At ‘‘e’’, depolarization rate is comparable to th,A. Consequently, activation of the A-current becomes large before inactivation becomes small (hA40 during depolarization) and the A-current increases. As an outward current, it dampens the peak, and— Z+ continues to decrease until it reaches a local minimum around 4 Hz (‘‘f’’ in Fig. 3B1). At this frequency, the duration of the hyperpolarizing window becomes short relative to th,A (hAo1 during hyperpolarization). As a result, the dampening effect of the A-current on the peak is incrementally smaller, and — Z+ increases. At larger frequencies (beyond point ‘‘g’’), the A-current is no longer active (hA0 during hyperpolarization):— Z+ and— Z merge, both decreasing together as frequency increases, due to the passive properties of the cell. As an initial condition, let fintrinsicfcross. In the intact network, the PD neuron is subject to synaptic inputs and fnatural may be different than fintrinsic. For simplicity, assume that inhibitory and excitatory inputs decrease and increase the cycle frequency, respectively (note that this is not necessarily always the case, see for example [6]). Taking account of such inputs, suppose that some external perturbation (for example, an increase in the strength of excitatory inputs, or injection of positive current in the cell) increases the cycle frequency such that fnatural4fcross. In this frequency regime, we saw that the PD neuron is —). relatively more responsive to negative inputs ( — Z+oZ Hence, as a result of the increased weight of inhibitory inputs, the rhythm slows down back to fcross. Similarly, if an external perturbation decreases the cycle frequency such that fnaturalofcross the PD neuron becomes relatively more responsive to positive inputs ( — Z +4 — Z), and the rhythm accelerates back to fcross. In this sense, we propose that fcross acts as a fixed point for the frequency of the pacemaker rhythm when it is subject to synaptic influences. We reasoned that if fnatural and fcross are related through this proposed mechanism, modification of fintrinsic should affect fcross; this, in turn, should affect fnatural. To explore this hypothesis, we changed some parameter such that fintrinsic was varied. Then, we examined how this manipulation changed the excitability profile and in particular the fcross value. We repeated these simulations for several different values of fintrinsic. We then plotted fcross as function of the corresponding fintrinsic. There are several possibilities to vary fintrinsic: (1) by increasing the interburst duration in PD; (2) by increasing the burst duration in PD; or (3) by increasing both while maintaining the duty cycle. In our model, the best parameter to vary fintrinsic was the time constant of Tcurrent inactivation, th,T(V), because its voltage dependence enabled us to choose which portion of the cycle to change, without significantly affecting the rest of the cycle:
Fig. 4. Relationship between fcross and fintrinsic in model: (A) three methods of changing the rhythm and (B) for each case, fcross is plotted as function of the corresponding fintrinsic. Dotted line: x ¼ y.
for example, increasing th,T only at low membrane potentials (lower than burst threshold) mainly increased the interburst duration; increasing th,T only at high membrane potentials mainly affected the burst duration (Fig. 4A). We now examine how changing fintrinsic by any one of these 3 possibilities affected fcross. When fintrinsic was decreased by increasing th,T at low membrane potentials (hence, mainly increasing the interburst duration), the Tcurrent deinactivated more slowly during hyperpolarization and the enhancing effect of the T-current was restricted to smaller frequencies. In the excitability profile, this resulted in a smaller fcross. Hence, fintrinsic and fcross were almost directly correlated (black squares, Fig. 4B). The two other ways of changing the rhythm did not yield such a correlation.
4. Discussion We demonstrated that the interaction between a rapid Acurrent a slower T-current could produce an excitability profile where positive and negative inputs are differentially
ARTICLE IN PRESS I. Saraf-Sinik, Y. Manor / Neurocomputing 69 (2006) 1103–1107
preferred at small and large frequencies, respectively. Our simulation results indicate that the frequency at which positive and negative inputs are equally effective may be related to the intrinsic frequency of a pacemaker neuron, provided that pacemaker burst duration is fixed. Interestingly, experimental manipulations that decrease the pyloric frequency mostly increase the interburst duration of pacemakers, with minimal effects on burst duration. Our working hypothesis is that both inhibitory and excitatory inputs are necessary to stabilize the pyloric rhythm around some internally determined fixed point. According to this hypothesis, removal of inhibitory sources is expected to increase the sensitivity of the pacemaker kernel to any treatment that speeds up the pyloric rhythm: despite the increased responsiveness of the pacemaker neuron to inhibitory inputs, the rhythm does not converge back to its attractor because such inputs are missing. Likewise, removal of excitatory sources is expected to increase the sensitivity of the pacemaker kernel to any treatment that slows down the pyloric rhythm. We will examine these predictions with the aid of the dynamic clamp method.
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[3] E. Marder, R.L. Calabrese, Principles of rhythmic motor pattern generation, Physiol. Rev. 76 (1996) 687–717. [4] J.P. Miller, A. Selverston, Rapid killing of single neurons by irradiation of intracellularly injected dye, Science 206 (1979) 702–704. [5] M.P. Nusbaum, M.P. Beenhakker, A small-systems approach to motor pattern generation, Nature 417 (2002) 343–350. [6] A.A. Prinz, V. Thirumalai, E. Marder, The functional consequences of changes in the strength and duration of synaptic inputs to oscillatory neurons, J. Neurosci. 23 (2003) 943–954. [7] M.J. Richardson, N. Brunel, V. Hakim, From subthreshold to firingrate resonance, J. Neurophysiol. 89 (2003) 2538–2554. [8] A.M. Swensen, E. Marder, Multiple peptides converge to activate the same voltage-dependent current in a central pattern-generating circuit, J. Neurosci. 20 (2000) 6752–6759. [9] A.L. Weaver, S.L. Hooper, Follower neurons in lobster (Panulirus interruptus) pyloric network regulate pacemaker period in complementary ways, J. Neurophysiol. 89 (2003) 1327–1338. Inbar Saraf- Sinik is a graduate student in the Life Sciences department of the Ben-Gurion University, Beer Sheva, Israel. She studies different mechanisms of frequency regulation in central pattern generators such as the pyloric rhythm of crustraceans. In her work, she combines electrophysiology with computational modeling.
Acknowledgments This research was supported by BSF 2001-039. References [1] I. Greenberg, Y. Manor, Synaptic depression in conjunction with A-current channels promote phase constancy in a rhythmic network, J. Neurophysiol. 93 (2005) 656–677. [2] B. Hutcheon, Y. Yarom, Resonance, oscillation and the intrinsic frequency preferences of neurons, Trends Neurosci. 23 (2000) 216–222.
Yair Manor is an associate professor in the Life Sciences department of the Ben-Gurion University, Beer Sheva. His main interest lies in mechanisms and regulation of rhythmic activity and the neuromodulation of small neuronal networks.
Neurocomputing 69 (2006) 1103–1107 www.elsevier.com/locate/neucom
Role of frequency-dependent weighing of inputs on frequency regulation of a pacemaker-driven rhythm Inbar Saraf-Sinik, Yair Manor Life Sciences Department, Ben-Gurion University, Beer Sheva 84105, Israel Available online 3 February 2006
Abstract We examined the frequency-dependent efficiency of positive and negative inputs onto pacemaker neurons of the pyloric rhythm. When neuromodulation to the pyloric circuit was blocked, pyloric neurons became quiescent. In the pacemaker neurons, we injected series of sinusoidal currents of different frequencies. At slow frequencies, voltage peak amplitudes were significantly larger than troughs. This trend reversed at higher frequencies. A computational model suggests that rapid A- and slower T-type currents can produce this excitability profile, and that a frequency dependence preference of negative and positive inputs impinging onto pacemaker neurons can play an important role in homeostatic stabilization of the rhythm. r 2006 Elsevier B.V. All rights reserved. Keywords: Pacemaker; Frequency-dependent excitability; Oscillation; Stomatogastric ganglion; Pyloric rhythm
1. Introduction Many rhythmic activities are mediated by central pattern generators (CPGs) [3]. A key question is what properties endow a CPG with regularity and flexibility at the same time. It is intuitive to accept that some types of external neuromodulation could control the CPG frequency by differentially strengthening or weakening inhibitory versus excitatory inputs impinging onto CPG neurons. One source of complication to consider is the frequency-dependent excitability of these neurons. In sub-threshold regimes, the voltage responses to positive or negative currents of the same amplitude are similar, hence a frequency-dependent transfer function (impedance) is a reliable tool to quantify the frequency-dependent excitability properties [2,7]. However, in biological neurons voltage responses to currents of opposite signs are not necessarily symmetrical, especially when these currents are large. Hence, a measure of frequency-dependent excitability that distinguishes between inhibitory and excitatory inputs may be more informative. Corresponding author.
E-mail address: [email protected] (Y. Manor). 0925-2312/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2005.12.054
The well-studied pyloric rhythm is an example of a CPG which pattern is affected by inhibitory and excitatory inputs [5]. When descending neuromodulation to the pyloric network is intact, this CPG generally produces a stable oscillation (1 Hz, hereafter referred to as fnatural). Occasionally, however, the rhythm frequency can vary between 0.3 and 2.5 Hz. The oscillations are generated by a group of electrically coupled conditional oscillators, the anterior burster (AB) neuron and the two pyloric dilator (PD) neurons. The oscillatory activity is propagated to follower neurons via chemical inhibitory and electrical synapses (see diagram in Fig. 2). Two followers, the lateral pyloric (LP) neuron and the ventricular dilator (VD) neuron, provide negative and positive inputs back to the pacemaker neurons. Within a range of cycle periods, these two neurons may regulate the pyloric frequency in complementary ways [9]. In this work we developed a measure of frequencydependent excitability that distinguishes between positive and negative inputs. Focusing on the PD neuron, we examine the possibility that frequency-dependent preference of negative and positive inputs impinging on pyloric pacemaker neurons contributes to the stabilization of the pyloric rhythm.
ARTICLE IN PRESS 1104
I. Saraf-Sinik, Y. Manor / Neurocomputing 69 (2006) 1103–1107
2. Methods
2.3. Modeling
2.1. Electrophysiology All experiments were done on the isolated stomatogastric nervous system of the lobster Homarus americanus. Intracellular recordings of PD neurons were done with sharp microelectrodes. PD neurons were isolated from their neighbors by blocking chemical synapses with bath application of picrotoxin (5 106 M), and killing electrically coupled cells with photoinactivation techniques [4]. Descending neuromodulation was impaired by blocking all electrical activity in the stomatogastric nerve (stn), usually by bath application of a solution containing tetrodotoxin (107 M). Following this procedure, the pyloric rhythm was abolished and all pyloric neurons became quiescent.
The network model consisted of three interconnected cells: an auto-rhythmic cell (representing the PD neuron) and two follower cells (representing the VD and LP neurons). The PD model was coupled to the VD model with a rectifying electrical synapse, and to the LP model via reciprocal inhibition. All three cells were implemented as conductance-based models, comprising leak, transient sodium, delayed rectifier, T-like calcium and A-like potassium currents. The PD cell also included a ‘‘proctolin current’’ (a ligand-activated voltage-dependent inward current), which represented the effect of neuromodulation [8]. Time-dependent variables were numerically integrated with the fourth-order Runge–Kutta method, as described in [1].
2.2. Construction of the frequency-dependent excitability profile
3. Results
In a quiescent PD neuron, we injected a ‘‘sweep’’ current waveform (amplitude: 73 nA). The sweep waveform consisted of a consecutive series of 13 sinusoidal waves of discrete frequencies (f 1 ¼ 0:1, f 2 ¼ 0:2; . . ., f 10 ¼ 1:0, f 11 ¼ 1:5; . . ., f13 ¼ 3.0 Hz). At any frequency fi, a sine wave was injected for 2 (io8) or 4 (iX8) cycles. To eliminate transient effects and allow a smooth transition between any two discrete frequencies fi and fi+1, following the 2 (or 4) cycles of fi a single transition cycle was injected. The frequency of this transition cycle was continuously changed from fi to fi+1. Fig. 1 shows a portion of the sweep waveform current. For each discrete frequency fi, on the voltage response (Fig. 1A) we measured the peak and trough amplitudes of the last cycle (open symbols). These values were then divided by current amplitude, and normalized with respect to input resistance. This procedure yielded the two functions — Z+(f) (for peaks) and — Z(f) (for troughs), that together defined the excitability profile of the cell.
Fig. 1. The sweep current waveform stimulus and voltage response of the PD neuron.
In the intact network, biological PD neurons showed a typical oscillatory activity (Fig. 2). Following synaptic isolation of the PD neuron, the rhythm in the PD neuron slowed down (from 1.0170.24 to 0.76 Hz70.02, n ¼ 2). We hereafter refer to the cycle frequency of the rhythm produced by the isolated PD neuron as fintrinsic ( ¼ 1/ Pintrinsic). When neuromodulation was blocked, oscillations disappeared. Fig. 2B reproduces these experimental observations in a network model. We tuned the parameters of the model such that, in the intact network, the three cells produced a 1 Hz rhythm. When feedback connections from the followers were disabled, the rhythm slowed down to 0.75 Hz. Bath application of tetrodotoxin, which blocked neuromodulation, was modeled by setting sodium and proctolin conductances in the PD model to 0. Under this condition, oscillations of network model stopped. The excitability profiles of 37 biological PD neurons were measured as described in Methods. Fig. 3A2 is an example of a PD voltage trace in response to intracellular injection of a current sweep waveform. The average excitability profile (Mean7SE, n ¼ 37) is shown in Fig. 3A1. Interestingly, within the stomatogastric ganglion we observed similar excitability profiles in the AB neuron only. In theory, the enhanced responsiveness at slow frequencies could be due to a slowly inactivating inward current (for example, a T-current), whereas a rapidly inactivating outward current (for example, an A-current) could explain the decreased responsiveness at high frequencies. In fact, in our experiments we observed that the characteristic frequency-dependence excitability profiles of biological PD neurons were disrupted in the presence of A- or Tcurrent blockers (data not shown). Hence, we reasoned that A- and T-current could be responsible for this characteristic excitability profile. Therefore, throughout this work the PD cell was modeled with a rapidly inactivating A-current (small th,A relative to Pintrinsic),
ARTICLE IN PRESS I. Saraf-Sinik, Y. Manor / Neurocomputing 69 (2006) 1103–1107
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Fig. 2. The pyloric rhythm: (A) biological and (B) computational model.
Fig. 3. The frequency-dependent excitability profile of a PD neuron: (A) biological and (B) model.
and a slowly inactivating T-current (large th,T relative to Pintrinsic). Fig. 3B2 shows the voltage response of an isolated PD model cell (simulated with a blocked neuromodulation, and with no feedback connections from follower neurons) to current injection of a sweep waveform. Here we extended the range of frequencies, starting from f1 ¼ 0.01 to f72 ¼ 50 Hz. Although frequencieso0.1 Hz and 43 Hz are not physiological, examination of the PD neuron responsiveness in this extended range of frequencies is informative, as detailed below. In the physiological range (o3 Hz), except for small discrepancies the excitability profile was similar to the one obtained in the biological PD neurons. There was essentially no effect of frequency on the troughs of PD responses (— Z): during the hyperpolarization phases the cell essentially acted as a low-pass filter with a cut-off frequency around 3 Hz. The effect of frequency on the peaks ( — Z+) can be explained by examining the effects
of frequency on the slow T-current and faster A-current. Let hA and hT represent the gating variables of A- and Tcurrent inactivation, respectively. At slow frequencies (‘‘a’’ in Fig. 3B1) depolarization is slow relative to th,A and th,T, and in both cases the currents inactivate (hA, hT-0) before they can activate. Hence, neither current is active. As f increases (‘‘b’’ in Fig. 3B1), depolarization rate is still slow relative to th,A, hence the A-current does not affect the peak (hA-0 during depolarization). However, depolarization rate is already comparable to th,T, and T-current activates before it inactivates (hT40 during depolarization). As a result, the T-current becomes larger and, as an inward current, it enhances the peak of the oscillation: — Z+ + increases. At f ¼ 0.3 Hz (‘‘c’’ in Fig. 3B1), — Z reaches a local maximum. A larger frequency means not only a faster depolarization, but also a shorter duration of the hyperpolarizing window: beyond ‘‘c’’, the hyperpolarizing window becomes short relative to th,T. The deinactivation of the T-current is incrementally smaller (hTo1 during
ARTICLE IN PRESS 1106
I. Saraf-Sinik, Y. Manor / Neurocomputing 69 (2006) 1103–1107
hyperpolarization), thereby moderating its effect on the peak, and — Z+ decreases. This trend continues until the Tcurrent can no longer deinactivate (hT0 during hyperpolarization): beyond this frequency, the T-current does not contribute to the peak. This occurs around f ¼ 0.7 Hz (‘‘d’’ in Fig. 3B1), where — Z+ intersects with — Z. At ‘‘e’’, depolarization rate is comparable to th,A. Consequently, activation of the A-current becomes large before inactivation becomes small (hA40 during depolarization) and the A-current increases. As an outward current, it dampens the peak, and— Z+ continues to decrease until it reaches a local minimum around 4 Hz (‘‘f’’ in Fig. 3B1). At this frequency, the duration of the hyperpolarizing window becomes short relative to th,A (hAo1 during hyperpolarization). As a result, the dampening effect of the A-current on the peak is incrementally smaller, and — Z+ increases. At larger frequencies (beyond point ‘‘g’’), the A-current is no longer active (hA0 during hyperpolarization):— Z+ and— Z merge, both decreasing together as frequency increases, due to the passive properties of the cell. As an initial condition, let fintrinsicfcross. In the intact network, the PD neuron is subject to synaptic inputs and fnatural may be different than fintrinsic. For simplicity, assume that inhibitory and excitatory inputs decrease and increase the cycle frequency, respectively (note that this is not necessarily always the case, see for example [6]). Taking account of such inputs, suppose that some external perturbation (for example, an increase in the strength of excitatory inputs, or injection of positive current in the cell) increases the cycle frequency such that fnatural4fcross. In this frequency regime, we saw that the PD neuron is —). relatively more responsive to negative inputs ( — Z+oZ Hence, as a result of the increased weight of inhibitory inputs, the rhythm slows down back to fcross. Similarly, if an external perturbation decreases the cycle frequency such that fnaturalofcross the PD neuron becomes relatively more responsive to positive inputs ( — Z +4 — Z), and the rhythm accelerates back to fcross. In this sense, we propose that fcross acts as a fixed point for the frequency of the pacemaker rhythm when it is subject to synaptic influences. We reasoned that if fnatural and fcross are related through this proposed mechanism, modification of fintrinsic should affect fcross; this, in turn, should affect fnatural. To explore this hypothesis, we changed some parameter such that fintrinsic was varied. Then, we examined how this manipulation changed the excitability profile and in particular the fcross value. We repeated these simulations for several different values of fintrinsic. We then plotted fcross as function of the corresponding fintrinsic. There are several possibilities to vary fintrinsic: (1) by increasing the interburst duration in PD; (2) by increasing the burst duration in PD; or (3) by increasing both while maintaining the duty cycle. In our model, the best parameter to vary fintrinsic was the time constant of Tcurrent inactivation, th,T(V), because its voltage dependence enabled us to choose which portion of the cycle to change, without significantly affecting the rest of the cycle:
Fig. 4. Relationship between fcross and fintrinsic in model: (A) three methods of changing the rhythm and (B) for each case, fcross is plotted as function of the corresponding fintrinsic. Dotted line: x ¼ y.
for example, increasing th,T only at low membrane potentials (lower than burst threshold) mainly increased the interburst duration; increasing th,T only at high membrane potentials mainly affected the burst duration (Fig. 4A). We now examine how changing fintrinsic by any one of these 3 possibilities affected fcross. When fintrinsic was decreased by increasing th,T at low membrane potentials (hence, mainly increasing the interburst duration), the Tcurrent deinactivated more slowly during hyperpolarization and the enhancing effect of the T-current was restricted to smaller frequencies. In the excitability profile, this resulted in a smaller fcross. Hence, fintrinsic and fcross were almost directly correlated (black squares, Fig. 4B). The two other ways of changing the rhythm did not yield such a correlation.
4. Discussion We demonstrated that the interaction between a rapid Acurrent a slower T-current could produce an excitability profile where positive and negative inputs are differentially
ARTICLE IN PRESS I. Saraf-Sinik, Y. Manor / Neurocomputing 69 (2006) 1103–1107
preferred at small and large frequencies, respectively. Our simulation results indicate that the frequency at which positive and negative inputs are equally effective may be related to the intrinsic frequency of a pacemaker neuron, provided that pacemaker burst duration is fixed. Interestingly, experimental manipulations that decrease the pyloric frequency mostly increase the interburst duration of pacemakers, with minimal effects on burst duration. Our working hypothesis is that both inhibitory and excitatory inputs are necessary to stabilize the pyloric rhythm around some internally determined fixed point. According to this hypothesis, removal of inhibitory sources is expected to increase the sensitivity of the pacemaker kernel to any treatment that speeds up the pyloric rhythm: despite the increased responsiveness of the pacemaker neuron to inhibitory inputs, the rhythm does not converge back to its attractor because such inputs are missing. Likewise, removal of excitatory sources is expected to increase the sensitivity of the pacemaker kernel to any treatment that slows down the pyloric rhythm. We will examine these predictions with the aid of the dynamic clamp method.
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[3] E. Marder, R.L. Calabrese, Principles of rhythmic motor pattern generation, Physiol. Rev. 76 (1996) 687–717. [4] J.P. Miller, A. Selverston, Rapid killing of single neurons by irradiation of intracellularly injected dye, Science 206 (1979) 702–704. [5] M.P. Nusbaum, M.P. Beenhakker, A small-systems approach to motor pattern generation, Nature 417 (2002) 343–350. [6] A.A. Prinz, V. Thirumalai, E. Marder, The functional consequences of changes in the strength and duration of synaptic inputs to oscillatory neurons, J. Neurosci. 23 (2003) 943–954. [7] M.J. Richardson, N. Brunel, V. Hakim, From subthreshold to firingrate resonance, J. Neurophysiol. 89 (2003) 2538–2554. [8] A.M. Swensen, E. Marder, Multiple peptides converge to activate the same voltage-dependent current in a central pattern-generating circuit, J. Neurosci. 20 (2000) 6752–6759. [9] A.L. Weaver, S.L. Hooper, Follower neurons in lobster (Panulirus interruptus) pyloric network regulate pacemaker period in complementary ways, J. Neurophysiol. 89 (2003) 1327–1338. Inbar Saraf- Sinik is a graduate student in the Life Sciences department of the Ben-Gurion University, Beer Sheva, Israel. She studies different mechanisms of frequency regulation in central pattern generators such as the pyloric rhythm of crustraceans. In her work, she combines electrophysiology with computational modeling.
Acknowledgments This research was supported by BSF 2001-039. References [1] I. Greenberg, Y. Manor, Synaptic depression in conjunction with A-current channels promote phase constancy in a rhythmic network, J. Neurophysiol. 93 (2005) 656–677. [2] B. Hutcheon, Y. Yarom, Resonance, oscillation and the intrinsic frequency preferences of neurons, Trends Neurosci. 23 (2000) 216–222.
Yair Manor is an associate professor in the Life Sciences department of the Ben-Gurion University, Beer Sheva. His main interest lies in mechanisms and regulation of rhythmic activity and the neuromodulation of small neuronal networks.