Experimental evidence for conditional backpropagation

It was shown early on [125] that active backpropagation of Na+ spikes into the apical dendrite is an integral part of high-frequency burst generation by pyramidal cells, similar to what has been described for several other systems [50]. Spikes are initiated at the soma or axon hillock and travel back into the apical dendrite up to the first major branch points (^200 mm). Membrane depolarization and repolarization in the dendrite are slower than in the soma and therefore dendritic spikes are longer in duration than somatic ones. A fast afterhyperpolarization (AHP) of the somatic membrane increases the potential difference between the soma and the still depolarized dendrite and leads to a sizable amount of current being sourced back into the soma where it supports a depolarizing afterpotential (DAP; Figure 8.9). In the course of a burst, somatic DAP amplitude is potentiated because of frequency-dependent broadening of dendritic spikes. Consecutive DAPs sum up and cause the frequency of somatic spike generation to increase. Eventually, the DAP itself will reach threshold for spike initiation and a high-frequency somatic spike doublet will be generated (ISI typically < 6 ms). Since the refractory period of the apical dendrite is longer (~4.5 ms) than that of the soma 2 ms), the dendrite does not support active back-propagation of the second spike of the doublet, and the corresponding DAP at the soma fails allowing the AHP to terminate the burst (Figure 8.9b). This mechanism of burst generation and termination has been termed conditional backpropagation [75], because backpropagation is essential for burst production, and it is conditional on sufficiently low spike frequencies. When spike frequency exceeds the dendritic refractory period, backpropagation fails and the burst is terminated.

A number of cellular components of the burst mechanism have been identified. Na+ channels are distributed in a punctate manner along the proximal 200 mm of the apical dendrite consistent with the finding that active backpropagation of TTX-sensitive spikes terminates at about this distance from the soma [125]. A candidate mechanism for the broadening of dendritic spikes is cumulative inactivation of a dendritic K+-conductance [75]. The inactivation would slow the repolarization of the dendritic membrane potential in a spike-frequency-dependent manner, thus increasing the amplitude of the somatic DAP. A likely candidate for this current is the Apteronotid homologue of the mammalian Kv3.3 K+-channel (AptKv3.3), which is extensively distributed along the entire axis of pyramidal cells [96, 97]. Local blockade of dendritic AptKv3.3 led to slowing of spike repolarization and increase in somatic DAP with a time-course similar to that of a regular burst. This manipulation also lowered the threshold for burst discharge evoked by current injection into the soma [97]. Therefore, it seems likely that this high-voltage-activated K+ channel is either directly involved in the mechanism of burst discharge or at the very least can modulate the threshold for burst generation [93]. Another contribution to the potentiation of the somatic DAP in the course of a burst comes from a persistent Na+ current which is activated by the increasing dendritic spike duration [34]. In contrast to other systems (for review see [56]), Ca2+ currents or Ca2+ -dependent K+ currents do not appear to be necessary for burst generation [34, 75, 93].

The detailed knowledge of pyramidal cell morphology, the organization of primary sensory and feedback input, and of the conductances shaping burst firing in vitro, makes pyramidal cells ideally suited for detailed modeling of the mechanism underlying burst firing. This mechanism differs in interesting ways from burst generation as described in several other systems. One obvious peculiarity of ELL pyramidal cell bursts is that ISI duration decreases in the course of a burst (Figure 8.9b), a phenomenon that has not been described in any other system so far. In vivo, however, this ISI pattern can be observed only rarely (Krahe, unpublished observations). With natural synaptic input, other factors like inhibition and the interplay between affer ent and feedback input may also shape the bursts and contribute to their termination. Furthermore, the basilar dendrites of E-units warrant closer investigation since they have been shown to be equipped with Na+ channels as well as AptKv3.3 K+ channels, and might thus also support backpropagation and bursting in a way similar to the apical dendrite [96, 97,125] (see also [100] for similar conclusions in neocortical pyramidal neurons).

8.5.2 Multicompartmental model of pyramidal cell bursts

Based on the detailed spatial reconstruction of a dye-filled E-type pyramidal cell [17], Doiron et al. [33, 34] developed a multicompartmental model that successfully reproduces burst firing as it is observed in vitro (Figure 8.9). The main goal of these studies was to identify the components of the burst mechanism that underlie dendritic spike broadening and somatic DAP potentiation since those are responsible for the progressive decrease in ISI duration and eventual burst termination. A key feature of the model was the presence of fast Na+ and K+ currents in both somatic and dendritic compartments, to account for Na+ action potential generation and backpropagation (Figure 8.10a). In order to achieve the narrow somatic and broader dendritic spike shapes (see 5.5.1), the time constants of the active conductances in the dendrite had to be increased relative to the soma. This also yielded a relatively longer refractory period for the dendritic spike compared to the somatic one.

While the core model outlined above reproduced key features of the somatic and dendritic response, it failed to generate spike bursts. Doiron et al. [33] were able to exclude a number of potential burst mechanisms described for other systems: Ca2+-or voltage-dependent slowly activating K+ channels, slow inactivation of the dendritic Na+ channel, and slow activation of the persistent Na+ current. Finally, modification of the dendritic delayed rectifier channel yielded burst properties corresponding to the in vitro findings: A low-threshold slow inactivation of the K+ conductance led to dendritic spike broadening in the course of a burst and to a corresponding increase in the DAP amplitude, which eventually triggered a doublet, leading to dendritic spike failure and burst termination due to the AHP. Whereas slow activation of the persistent Na+ current proved insufficient to elicit proper bursting, it was recently shown to be an important component of the DAP potentiation [34]. It is activated by the broadening of dendritic spikes and boosts the sub-threshold depolarization of the somatic membrane. Thereby it largely determines the time it takes to reach threshold for doublet firing. Since the doublet terminates the burst, the persistent Na+ current thus controls burst duration. With the interburst period being largely fixed by the duration of the AHP, the persistent Na+ current also determines burst oscillation period [34]. Since it can be activated by descending feedback to the apical dendrites [15, 17], this provides a potential mechanism for controlling burst firing depending on behavioral context.

To summarize, the key features of the pyramidal cell burst mechanism are i) a dendritic Na+ conductance that supports active backpropagation of spikes into the dendrite and that feeds the somatic DAP, ii) a slow cumulative inactivation of a delayed rectifier current which leads to dendritic spike broadening in the course of a

Figure 8.9

Summary of the mechanism underlying high-frequency burst generation in pyramidal cells in vitro. a) Schematic diagram of a pyramidal cell with a narrow spike recorded in the soma (1). The somatic spike is actively propagated back into the apical dendrite where a much broader version of the same spike can be recorded (2). Current sourcing from the dendrite back into the soma causes a DAP (3). b) Top: Oscillatory burst discharge recorded in the soma of a pyramidal cell with 0.74 nA depolarizing current injection. Middle and bottom: Somatic and dendritic spike burst recorded separately in two different cells. The time scales are adjusted to allow alignment of spikes. Somatic spikes are truncated. As evident from the dendritic recording, spike repolarization slows down in the course of a burst allowing the DAP at the soma to potentiate. Eventually, the DAP reaches threshold and causes a high-frequency spike doublet. Since the dendritic refractory period is longer than the somatic one, the dendrite cannot support active propagation of the second spike of the doublet. The DAP fails and allows the afterhyperpolarization (AHP) to terminate the spike burst. (a) adapted from [75], (b) adapted from [34].

burst, thus potentiating the somatic DAP, iii) a shorter refractory period for somatic spikes compared to dendritic ones renders backpropagation conditional on the instantaneous firing rate, iv) the rate of the DAP potentiation, which is part of a positive feedback loop in which dendritic spike broadening activates a persistent Na+ current, which further boosts depolarization. The slow dynamics of the persistent Na+ current largely control burst duration and burst frequency.

8.5.3 Reduced models of burst firing

Detailed biophysical models are powerful tools for probing the understanding of cellular mechanisms at a microscopic scale. However, they are computationally too complex for modeling of large networks or for analyzing the behavior of single cells from a dynamical systems perspective. Having understood the key mechanisms, it is often possible to reduce a detailed biophysical model to its essential components and then apply dynamical systems analysis to the lower-dimensional model [101]. The multi-compartmental model described above has undergone two such reductions, first to a two-compartment model, termed a ghostburster for reasons explained in more detail below [32], and then to an even simpler two-variable delay-differential-equation model [71].

To model the generation of the somatic DAP, only a somatic and one dendritic compartment representing the entire apical dendritic tree are needed (Figure 8.11a) [32]. Soma and dendrite were equipped with fast Na+ channels, delayed rectifier K+ currents, and passive leak current. Current flow between the compartments followed simple electrotonic gradients determined by the coupling coefficient between the two compartments, scaled by the ratio of somatic to total model surface (see also [64, 80, 129]). Thus, the entire system was described by only six nonlinear differential equations using modified Hodgkin/Huxley kinetics [54]. To achieve the relatively longer refractory period of the dendrite [75], the time constant of dendritic Na+ inactivation was chosen to be longer than somatic Na+ inactivation and somatic K+ activation. The key element for the burst mechanism was the introduction of a slow inactivation variable for the dendritic delayed rectifier current, whose time constant was set to about 5 times slower than the mechanisms of spike generation. In this configuration, the two-compartment model reliably reproduced the potentiation of the somatic DAP, which eventually triggers the firing of a spike doublet, the burst termination due to failure of backpropagation, and the rapid onset of the AHP [32] (see Figure 8.9b).

To study the burst dynamics, the ghostburster model was treated as a fast-slow burster [57, 101], separating it into a fast subsystem representing all variables related to spike generation, and a slow subsystem representing the dendritic K+ inactivation variable, pd. The fast subsystem could then be investigated using the slow variable as a bifurcation parameter. The dashed lines in Figure 8.11b show the quasi-static bifurcation diagram with maximum dendritic membrane voltage as a representative state variable of the fast subsystem, and pd as the slow subsystem. For constant values of pd > pdi, there exists a stable period-one solution. At pd = pd1 the fast subsystem transitions to a period-two limit cycle. This corresponds to intermittent doublet firing

Figure 8.10

Multi-compartmental model of burst generation. a) The model was based on the detailed reconstruction of a dye-filled E-type pyramidal cell [17]. The distribution of ionic channels along the neuron's axis is indicated in the insets. The detailed placement of Na+ and K+ channels in separate compartments of the proximal den-drite is shown on the left. b) The model reproduces the increasing firing frequency in the course of a burst with a doublet at the end and a burst AHP (top). The dendritic delayed-rectifier conductance, gDr,d, shows cumulative inactivation as the burst evolves (middle). The dendritic voltage-gated Na+ conductance, gNa,d, fails when the somatic ISI is within its refractory period (bottom). c) Summary graph showing the decrease in peak conductance of gDr,d and gNa,d as a function of spike number for the burst shown in b. Whereas gDr,d inactivates in a cumulative way, gNa,d decays much more gradually but is completely shut off by the high-frequency doublet. Adapted from [33].

Figure 8.10

Multi-compartmental model of burst generation. a) The model was based on the detailed reconstruction of a dye-filled E-type pyramidal cell [17]. The distribution of ionic channels along the neuron's axis is indicated in the insets. The detailed placement of Na+ and K+ channels in separate compartments of the proximal den-drite is shown on the left. b) The model reproduces the increasing firing frequency in the course of a burst with a doublet at the end and a burst AHP (top). The dendritic delayed-rectifier conductance, gDr,d, shows cumulative inactivation as the burst evolves (middle). The dendritic voltage-gated Na+ conductance, gNa,d, fails when the somatic ISI is within its refractory period (bottom). c) Summary graph showing the decrease in peak conductance of gDr,d and gNa,d as a function of spike number for the burst shown in b. Whereas gDr,d inactivates in a cumulative way, gNa,d decays much more gradually but is completely shut off by the high-frequency doublet. Adapted from [33].

lDr cj inactivation variable, p^

Figure 8.11

Two-compartment model of burst generation. a) Sketch of the somatic and dendritic compartments linked by an axial resistance. b) The dashed lines show the quasi-static bifurcation diagram with a representative of the fast subsystem, the maximum dendritic membrane voltage, as a function of the slow subsystem, the dendritic K+ inactivation variable, pd. Overlaid is a single burst trajectory (solid line; burst begins with the upwards pointing arrow on the right). Adapted from [32].

lDr cj inactivation variable, p^

Figure 8.11

Two-compartment model of burst generation. a) Sketch of the somatic and dendritic compartments linked by an axial resistance. b) The dashed lines show the quasi-static bifurcation diagram with a representative of the fast subsystem, the maximum dendritic membrane voltage, as a function of the slow subsystem, the dendritic K+ inactivation variable, pd. Overlaid is a single burst trajectory (solid line; burst begins with the upwards pointing arrow on the right). Adapted from [32].

with dendritic spike failure, since for pd < pd1 dendritic repolarization is sufficiently slow to cause very strong somatic DAPs capable of eliciting a second somatic spike after a small time interval (~3 ms). The overlaid burst trajectory (solid line) shows the beginning of the burst on the right side (upwards arrow). The maximum of the dendritic membrane voltage decreases for the second spike of the doublet (compare Figure 8.9b), which occurs at pd < Pdi. The short doublet ISI is followed by the long interburst ISI, the slow variable recovers until the next burst begins. Because pd is reinjected near an infinite-period bifurcation (saddle-node bifurcation of fixed points responsible for spike excitability), Doiron et al. [32] termed this burst mechanism ghostbursting (sensing the ghost of an infinite-period bifurcation [118]). Thus, the two-compartment model nicely explains the dynamics of pyramidal cell bursting observed in vitro by the interplay between fast spike-generating mechanisms and slow dendritic K+ -channel inactivation.

In a further reduction of the model, Laing and Longtin [71] replaced the six ordinary differential equation model by an integrate-and-fire model consisting of a set of two discontinuous delay-differential equations. An interesting aspect of this model is that it uses a discrete delay to mimic the ping-pong effect between soma and den-

drite. When a spike occurs, the somatic membrane potential is boosted by a variable amount but only if the preceding ISI was longer than the dendritic refractory period and only after a certain delay. The amount of somatic boosting depends on the firing history of the neuron. For long ISIs, it decays towards zero, for short ISIs it builds up.

Bifurcation analysis of both the ghostburster and the delay model revealed properties that contrast with other models of burst generation. When increasing amounts of current are injected into the soma, both reduced models move from quiescence for subthreshold current through a range of tonic periodic firing into irregular bursting (Figure 8.12) [32, 71]. The transition from quiescence to tonic firing is through a saddle-node bifurcation of fixed points after which the systems follow a stable limit cycle. The periodic attractor increases monotonically in frequency as current is increased. The fact that the models pass from quiescence to repetitive firing through a saddle-node bifurcation is characteristic of class I excitability [57, 101]. Accordingly, the neurons are able to fire at arbitrarily low rates close to the bifurcation, which is also observed when injecting small amounts of current into pyramidal cells in the slice preparation [75]. At higher current the models move through a saddle-node bifurcation of limit cycles after which they follow a chaotic attractor corresponding to burst firing. For very large input currents, the cells periodically discharge spike doublets (right of the dotted line in Figure 8.12 a, b). This progression from quiescence through periodic firing and bursting to periodic doublet discharge closely reproduces the behavior of pyramidal cells in the slice preparation [75]. Similar to the ghostburster model, the delay integrate-and-fire model also allows the generation of a wide gallery of bursts of different shapes indicating that pyramidal cells may be able to adjust burst duration and frequency depending on context.

The simplicity of the delay model also allowed examination of the effects of periodic forcing corresponding to injection of sinusoidal current at the soma. Depending on the frequency of sinusoidal forcing, the threshold for burst firing could be increased or decreased relative to the threshold in the unforced system. This finding suggests that depending on the frequency of amplitude modulations of the electric field, the threshold for burst firing of pyramidal cells might shift.

The most appealing aspect of the delay model is its simplicity and computational efficiency. Since the model captures the basic properties of burst firing described by the more elaborate ionic models [32, 33], it may be suitable for use in larger-scale models of electrosensory processing.

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