Figure 2: A) shows the functional anatomy of the beat timing circuit. Circles represent cell bodies and synaptic input sites, lines represents axons and processes, and squares represent specializations of the axons for spike initiation and synaptic input. HN cells are indexed by body side and ganglion number. HN1 and HN2 cells send axons down to G3 and G4, and make reciprocal inhibitory connections with HN cells on the same side in each ganglion. Each HN3 and HN4 cell makes a reciprocal spike-mediated and graded synaptic connection with its contralateral homologue. In addition, each HN3 cell sends an axon down to G4, and makes reciprocal inhibitory connections with the HN1 and HN2 cells on the same side. Red and yellow indicate the approximate activity phase of each cell, with red corresponding to ~0 degrees, and yellow to ~180 degrees. As described in the text, only the connections in G4 are modeled. B) shows the schematic of the modeled circuit. Since HN1 cell process in G4 fires in-phase with the ipsilateral HN2 process, and since the two cells make identical synaptic connections, they lumped together. Note that each HN1-2 synapse represents two synaptic connections, one for the HN1 cell and one for the HN2 cell. The synapses connecting the bilateral HN3 and HN4 cells also represent two connections, one spike-medated and the other graded.
The timing oscillation is dominated by the activity of the third and fourth pairs of heart interneurons. Reciprocally inhibitory synapses between these bilateral pairs of oscillator interneurons, combined with their ability to escape from inhibition and begin firing, pace the oscillation (Figure 3A). Each of these two reciprocally inhibitory heart interneuron pairs can be considered an elemental half-center oscillator. Activity within an elemental oscillator consists of alternating bursts of action potentials and quiescent intervals when the neuron is inhibited by its contralateral homologue (Figure 3A). These neurons are thus termed oscillator interneurons.
The first two pairs of heart interneurons act as coordinating interneurons,
serving to link these two elemental oscillators. Their pattern of synaptic
connections with the oscillator interneurons is illustrated in Figure 2
along with their rather unusual functional anatomy. These neurons initiate
spikes and receive synaptic inputs and make their synaptic outputs not
in their ganglion of origin but in ganglion 3 and 4. Under most circumstances
their primary site of spike initiation is in ganglion 4 where they receive
synaptic input from both ipsilateral HN(3) and HN(4) oscillator neurons.
For the purpose of modeling this synaptic diagram was simplified to reflect
the dominance of the G4 initiation sites of the coordinating interneurons
as shown in the inset of Figure 2. Activity in the entire timing network
is illustrated in the extracellular recordings of Figure 3B. Note that
the activity of the oscillator neurons on the same side is coordinated
with a small fixed phase difference and that the coordinating neurons fire
during the interval where both the oscillator neurons on the same side
are quiescent (i.e. they are being inhibited by their contralateral homologues
Figure 3: A shows a simultaneous intracellular recording of the elemental oscillator in G3. B shows extracellular recordings from three HN cells. The median spike of each burst is marked by a dot. Note that cells L,3 and L, 4 fire approximately in-phase, and that both of these cells fire approximately 180 degrees out of phase with cell L,2. There is a small phase difference between the bursts of cell L,3 and L,4: the phase difference is defined to be the delay divided by the cycle period.
Several ionic currents have been identified in single electrode voltage-clamp studies that contribute to the activity of oscillator heart interneurons (Calabrese et al. 1995). These include, in addition to the fast Na+ current that mediates spikes, two low-threshold Ca2+ currents [one rapidly inactivating (ICaF) and one slowly inactivating (ICaS)], three outward currents [a fast transient K+ current (IA) and two delayed rectifier-like K+ currents, one inactivating (IK1), and one persistent (IK2)], a hyperpolarization-activated inward current (Ih) (mixed Na+/K+, Erev=-0.020 V), a low-threshold persistent Na+ current (IP) and a leakage current (Il). The inhibition between oscillator interneurons consists of a graded component that is associated with the low-threshold Ca2+ currents and a spike-mediated component that appears to be mediated by a high-threshold Ca2+ current. Spike-mediated transmission varies in amplitude throughout a burst according to the baseline level of depolarization (Olsen and Calabrese 1996). Graded transmission wanes during a burst owing to the inactivation of low-threshold Ca2+ currents. Blockade of synaptic transmission with bicuculline leads to tonic activity in oscillator heart interneurons (Schmidt and Calabrese 1992) and Cs+, which specifically blocks Ih, leads to tonic activity or sporadic bursting (Angstadt and Calabrese 1989).
Much of this biophysical data has been incorporated into a detailed conductance-based model of an elemental (two-cell) oscillator (Nadim et al. 1995). The current generation of this model is implemented in this GENESIS (Bower and Beeman 1998) tutorial. The model cells consist of a single isopotential compartment and use standard Hodgkin-Huxley representations of each voltage-gated current. Synaptic transmission in the model is complex. A spike-triggered alpha-function is used to describe the postsynaptic conductance associated each action potential ,and the maximal conductance reached is a function of the past membrane potential to reflect the fact that spike-mediated transmission varies in amplitude throughout a burst according to the baseline level of depolarization. Graded synaptic transmission is represented by a synaptic transfer function, which relates postsynaptic conductance (the result of transmitter release) to presynaptic Ca2+ build-up and decline, via low-threshold Ca2+ currents and a Ca2+ removal mechanism respectively. The model is now in it third generation, having been upgraded each time by the incorporation of new data from experiments suggested by the previous generation of model (Olsen and Calabrese 1996). Free parameters in the model are the maximal conductance for each current (voltage-gated or synaptic). In the default or canonical version of the model presented here, the 's were adjusted to be close to the average observed experimentally and to produce and appropriate firing frequency during the burst phase of oscillation. The reversal potential, Eion, for each current was determined experimentally and they were considered fixed. Final selection of parameters to form a canonical model was dictated by model behavior under control conditions, passive response of the model to hyperpolarizing current pulses, and reaction of the model to current perturbations. The model cells were also required to fire tonically when all inhibition between them was blocked, as do the biological neurons (Schmidt and Calabrese 1992).
Much less is known about the intrinsic membrane properties of the coordinating neurons; because their unusual functional morphology prevents voltage clamp analysis from the soma. To construct the timing network model presented here the coordinating neuron were modeled similarly to the oscillator neurons except that they have a restricted set of voltage-gated currents. These currents ('s) were adjusted so that the neurons fire tonically at the experimentally observed frequency, whenever they are not inhibited by an oscillator neuron; when they are inhibited they are completely silent.
The steady-state activation/inactivation values and time-constant values
for all of the channels are shown graphically in Figure 5.
The actual value of the maximal conductance for each channel was taken to be a free parameter in the model, and was chosen so as to reproduce the frequency of oscillation typically observed in the animal. For the G3 and G4 cells, , , , , , , , , (all values are in nS).
The model cells representing the coordinating fibers are similar to those representing the oscillator neurons, except that the coordinating fibers have a higher leak reversal potential and only have three channels: Na, K1, and K2. For the G1 cells, , , , , and the G2 cells are identical except that (all values in nS). El= -0.040V for all of these cells.
Figure 5: Screen shots of the channel steady state activations and time constants as viewed in Neurokit. Two tau panels are shown for K1 (the delayed rectifier) and CaS (slow calcium) because the activation and inactivation values cannot be shown on the same scale. Note that the activation time course for the Na (fast sodium) is voltage indepdent--it has a constant value of 1.0e-04 seconds.
The equations describing the spike-mediated synapse and the graded synapse
are given in Figure 6. ESyn = -0.0625V. The
maximal synaptic conductances, like the maximal ionic channel conductances,
are taken as free parameters and are chosen so as to fit the data.
The values used in the regular model are the following: , , ,
(all units in nS).
Experiment 1: The relationship between ion channel conductances and single cell activity
Experiment 2: The relationship between intrinsic currents and two cell oscillations
Experiment 3: The relationship between the period of the circuit and the periods of its G3 and G4 components
First, many X11 window managers give the user the option to close a
window by clicking an "x" in the upper right corner of the window or a
square in the upper left corner. GENESIS is not equipped to deal
with these messages, so closing a window in this manner will cause the
entire program to quit. Every closeable window in this tutorial has
a "Close Window" or "close" button that should be used instead. Remember
the following figure:
Second, after editing a text field, one needs
to either hit return or click the label of the text field to register the
change. Sometimes the keys do not register, so it is a good idea
to hit return a couple of times instead of just once. If one clicks
the button, watch for it to change color as the mouse button is pressed
and released to ensure a good hit.
Finally, one can adjust the scale on graphs in GENESIS graphically by clicking on either the first or last label on an axis, and dragging it to a new location. For example, to change the scale for one of the Vm graphs to be from 0.010V ... -0.060V to 0.010mV ... -0.030mV, click on the label "-0.060" and drag up to the location of -0.030 on the axis. Then, release the mouse and the graph will redraw itself with the new scale. To move it back, grab the "-0.030" label, drag it off the graph until the label reads -0.060, and release the mouse.
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