Nonlinear interaction between shunting and adaptation controls a switch between integration and coincidence detection in pyramidal neurons

Abstract

The membrane conductance of a pyramidal neuron in vivo is substantially increased by background synaptic input. Increased membrane conductance, or shunting, does not simply reduce neuronal excitability. Recordings from hippocampal pyramidal neurons using dynamic clamp revealed that adaptation caused complete cessation of spiking in the high conductance state, whereas repetitive spiking could persist despite adaptation in the low conductance state. This behavior was reproduced in a phase plane model and was explained by a shunting-induced increase in voltage threshold. The increase in threshold allows greater activation of the M current (I(M)) at subthreshold potentials and reduces the minimum adaptation required to stabilize the system; in contrast, activation of the afterhyperpolarization current is unaffected by the increase in threshold and therefore remains unable to stop repetitive spiking. The nonlinear interaction between shunting and I(M) has other important consequences. First, timing of spikes elicited by brief stimuli is more precise when background spikes elicited by sustained input are prohibited, as occurs exclusively with I(M)-mediated adaptation in the high conductance state. Second, activation of I(M) at subthreshold potentials, which is increased in the high conductance state, hyperpolarizes average membrane potential away from voltage threshold, allowing only large, rapid fluctuations to reach threshold and elicit spikes. These results suggest that the shift from a low to high conductance state in a pyramidal neuron is accompanied by a switch from encoding time-averaged input with firing rate to encoding transient inputs with precisely timed spikes, in effect, switching the operational mode from integration to coincidence detection.

Figure 1.

Shunting modulates the outcome of adaptation in hippocampal CA1 pyramidal neurons. A, Sample responses to constant stimulation in a low and high conductance state (left and right traces, respectively). Horizontally aligned traces show responses with equivalent initial firing rate (based on reciprocal of first interspike interval). Stimulus intensity is indicated beside each trace. Despite increasing stimulus intensity to elicit the same initial firing rate in the low and high conductance states, the initial burst was shorter and repetitive spiking was completely abolished in the high conductance state. B, Based on the same cell illustrated in A, shunting caused a rightward shift in the f _initial –I _DC curve. C, Shunting also reduced the number of spikes in the initial burst. D, Whereas slow, repetitive spiking could persist after adaptation in the low conductance state, repetitive spiking was absent after adaptation in the high conductance state. E, Plotting initial firing rate against other response measures removes the direct effect of shunting on excitability and helps isolate how shunting influences the outcome of adaptation. For E and F, top graphs show data for cell illustrated in A and bottom graphs show cumulative data (n = 6 neurons, with 4–5 measurements per neuron for each condition). Lines in bottom graphs show linear regressions (low conductance, black line; high conductance, gray line). After accounting for variation in f _initial, there were significantly fewer spikes within the initial burst in the high conductance state compared with the low conductance state (p < 0.001, ANOVA). F, Steady-state firing rate was also significantly lower in the high conductance state compared with the low conductance state (p < 0.001, Kruskal–Wallis test).

Figure 2.

A modified Morris–Lecar model with I _M-mediated adaptation can reproduce the influence of shunting on repetitive spiking. A, Sample responses from the model neuron in the low and high conductance states (left and right traces, respectively). Horizontally aligned traces show responses with equivalent initial firing rate. Stimulus intensity is indicated beside each trace. As in real pyramidal neurons, the initial burst was shorter and repetitive spiking was absent in the high conductance state. B, Shunting caused a rightward shift in the f _initial –I _DC curve. C, It also reduced the number of spikes in the initial burst. D, Shunting also completely abolished steady-state spiking over the range of stimulus intensities tested here, although repetitive spiking could be achieved with much stronger stimulation (see Fig. 3 E). E, Bifurcation diagrams show that, regardless of membrane conductance, repetitive spiking was generated through a subcritical Hopf bifurcation in the w–V phase plane (see Figs. 3 B, 4 B). In this and all subsequent bifurcation diagrams, limit cycles are represented by plotting the maximum and minimum, during the limit cycle, of the variable plotted on the ordinate, which explains why these curves each have two branches. In the low conductance state, the bifurcation occurred at I _DC = I _∗ = 25.4 μA/cm² and V = V _∗ = −46.5 mV. Those values increased to I _∗ = 98.8 μA/cm² and V _∗ = −37.8 mV in the high conductance state. Bifurcation diagrams were generated with adaptation removed from the model neuron to characterize spike generation before adaptation; if adaptation were left in the model, the bifurcation would not occur until adaptation failed to prevent repetitive spiking (see Figs. 3 E, 4 E). F, Responses in the low and high conductance states are essentially the same in a model neuron with type I excitability (shown here) as in a model neuron with type II excitability (A).

Figure 3.

Effects of membrane conductance on repetitive spiking result from a nonlinear interaction between shunting and I _M-mediated adaptation. A, Sample responses in the model neuron for low and high conductance states (left and right traces, with I _DC = 40 and 110 μA/cm², respectively). Voltage (V), activation of the delayed rectifier current (w), and activation of adaptation (z) are plotted against time. Colored lines indicate times at which nullclines in B are calculated (see below). In the low conductance state, z decreases after the initial burst and only increases incrementally with each spike; in the high conductance state, z continues to increase after the initial burst in the absence of co-occurring spikes (arrow; see also arrows in B and D). B, The three-dimensional phase portrait (left panel of each set of three graphs) shows the evolution of each variable relative to the other two variables. The response is projected onto the w–V and z–V planes to assist visualization; the z–w plane adds no additional information and is not shown. Nullclines were calculated at three time points indicated by colored lines in A: before stimulation (red), at the onset of stimulation before adaptation develops (blue), and after adaptation (green). Values of w and z were frozen at the indicated values each time the V nullcline was calculated, and so on for each w and z nullcline. Nullclines were not replotted if they did not change from one time point to the next. Solid line represents the nullcline for the variable plotted on the abscissa; dashed line represents the nullcline for the variable plotted on the ordinate. Spike generation results from the interaction between V and w, whereas control of adaptation can be understood by the interaction between V and z. In both the w–V and z–V phase planes, excitatory I _DC shifts the V nullcline upward, whereas I _M shifts it downward. An increase in membrane conductance shifts the V nullcline to the right, which corresponds to an increase in V _∗ (purple line). The increase in V _∗ has two effects (see also C): first, it allows the same I _DC to cause a greater increase in z (based on where the blue V nullcline and z nullcline intersect); second, the resulting increase in activation of I _M stabilizes the system at a subthreshold potential (based on where the green V nullcline and z nullcline intersect). This last point can be seen most clearly in the insets, which show enlarged views of the phase space indicated by the orange arrowhead in the main graph: in the low conductance state, the adapted (green) V nullcline and z nullcline intersect to the right of V _∗, meaning that the intersection point is unstable (arrowhead labeled u) and repetitive spiking continues, whereas in the high conductance state, those nullclines intersect to the left of V _∗, meaning that the intersection point is stable (arrowhead labeled s) and I _M will maintain voltage below threshold. C, By reducing the model to two dimensions and treating z as a parameter instead of a variable, bifurcation diagrams show the minimum z required to stabilize the system (z _required) and stop repetitive firing; z _required = 0.12 and 0.08 for the low and high conductance states, respectively. Insets show activation curves for adaptation, which, compared with V _∗, show the maximum z achievable at subthreshold potentials (z _max, blue line); z _max = 0.09 and 0.37 for the low and high conductance states, respectively. The increase in z _max and the decrease in z _required associated with an increase in V _∗ confirms the observations made in B. Consequently, z _max < z _required in the low conductance state, whereas z _max > z _required (yellow region) in the high conductance state; the latter condition is required to stabilize voltage below threshold and terminate repetitive spiking. The same conclusion is found by superimposing the activation curve for adaptation (gray) onto the bifurcation diagram: in the low conductance state, the curve intersects the fixed point in a region of instability (dotted black line), meaning that repetitive firing persists; in the high conductance state, the curve intersects the fixed point in a stable region (solid black line), meaning that repetitive firing stops. D, Superimposing the z–V trajectory from B (green curve) onto the bifurcation diagrams from C illustrates how system tracks along the stable limit cycle at the onset of stimulation, with z increasing during each spike until it exceeds the bifurcation point. Behavior beyond this point depends on membrane conductance. In the low conductance state, spiking is transiently replaced by subthreshold voltage fluctuations, during which time z decreases (red arrow in inset) until another spike is generated (orange arrow), and so on. Conversely, in the high conductance state, z continues to increase after the termination of spiking. Stimulus intensity (I _DC) was equivalent in A–D. E, Bifurcation diagrams here show changes in z as I _DC is increased. In the low conductance state, there are four regions: subthreshold (region i), no repetitive firing after adaptation (ii), repetitive firing that is irregular (iiia), or regular (iiib) after adaptation. In the high conductance state, region iiia is missing. The absolute size of region ii increased in the high conductance state, but its size relative to region i remained almost unchanged at 56% in the high conductance state compared with 52% in the low conductance state. The relative size of region ii can, however, increase dramatically depending on adaptation parameters (see Results and Fig. 5).

Figure 4.

Adaptation mediated through I _AHP cannot prohibit repetitive firing, regardless of membrane conductance. A, Sample responses in the model neuron for low and high conductance states (left and right traces, with I _DC = 40 and 110 μA/cm², respectively). Voltage (V), activation of the delayed rectifier current (w), and activation of adaptation (z) are plotted against time. Colored lines indicate times at which nullclines in B are calculated. Unlike with I _M (see Fig. 3 A), z decreases between each spike and repetitive spiking continues regardless of shunting. B, Phase planes are represented here the same way as in Figure 3 B. The rightward shift in the z nullcline for I _AHP (compare with the z nullcline associated with I _M in Fig. 3 B) means that no I _AHP-mediated adaptation is induced or maintained at subthreshold potentials. Because of this, I _AHP is unable to shift the V nullcline down far enough to restabilize the system at V < V _∗ in either the low or high conductance state. Insets show enlarged views of the phase space indicated by the orange arrowhead in main graph. The adapted (green) V nullcline and w nullcline intersect to the right of V _∗, meaning that the intersection point is unstable (arrowhead labeled u) and repetitive spiking continues in both the low and high conductance states. In the z–V phase plane, the fixed point does not switch stability through a Hopf bifurcation (as described above for the w–V phase plane); instead, dynamics are altered by the destruction/creation of a fixed point through a saddle-node bifurcation (unlabeled arrowhead). C, As in Figure 3 C, the model was reduced to two dimensions by treating z as a parameter instead of a variable. Bifurcation diagrams show the minimum z required to stabilize the system (z _required) and stop repetitive firing; z _required = 0.16 and 0.010 for the low and high conductance states, respectively. Insets show activation curves for I _AHP, which, by comparing with V _∗, show the maximum z achievable at subthreshold potentials (z _max, blue line); z _max = 0.00 for both the low and high conductance states (i.e., adaptation cannot be induced without spike generation). Consequently, z _max < z _required regardless of membrane conductance, meaning that stabilization cannot be achieved at a subthreshold potential. This is similarly evident by the fact that activation curve of adaptation (gray), when superimposed on the bifurcation diagram, intersects the fixed point in a region of instability (dotted black line) for both the low and high conductance states. D, The z–V trajectory from B (green curve), when superimposed onto the bifurcation diagrams from C, shows that z decreases after the initial burst of spikes, allowing additional spikes to occur intermittently. This is the same behavior seen in the left panel of Figure 3 D and contrasts the behavior seen in the right panel of Figure 3 D. Stimulus intensity (I _DC) was equivalent in A–D. E, Bifurcation diagrams here show changes in z as I _DC is increased. Region ii (in which adaptation prevents repetitive firing) is completely absent in both the low and high conductance states.

Figure 5.

The range of bifurcation patterns seen in real pyramidal neurons can be reproduced in the model neuron with I _M-mediated adaptation. A, Graphs show stimulus intensity at which CA1 pyramidal neurons changed from a subthreshold response to an initial burst, followed by no repetitive firing (transition from region i → ii, solid line) and from no repetitive firing to repetitive firing (transition from region ii → iii, dashed line) plotted as cumulative probability across the six neurons tested. Labeled regions are defined by the median stimulus intensity at which a transition occurs. In the low conductance state (left), region ii was observed in four of six neurons, but its absence in two neurons suggests that region ii can be extremely narrow (see Results); for purposes of plotting, an intermediate transition through region ii was assumed for transitions from region i to iii. In the high conductance state (right), region ii was consistently wide and region iii was never observed over the range of stimulus intensities tested. B, In the model neuron, region ii can be arbitrarily narrow at one conductance level and dramatically widen as membrane conductance increases. For the bifurcation diagrams shown here, I _M-mediated adaptation was adjusted such that ḡ _M = 4 mS/cm² and γ = 2 mV. Repetitive firing predominated in the low conductance state (left), as indicated by the breadth of region iii relative to region ii, whereas repetitive firing was completely abolished in the high conductance state (right), as indicated by the absence of region iii and widening of region ii. This demonstration confirms that the variability between neurons can be accounted for by variation of model parameters within physiologically realistic limits and does not require a qualitative change in mechanism.

Figure 6.

Explanation and experimental confirmation of a shunting-induced increase in voltage threshold. A, The first two spikes elicited by step depolarization of the model neuron are shown for the low and high conductance states (left and right, respectively). Phases of the action potential are colored differently to help relate the voltage response plotted against time (top row) with the relationship between I _membrane and voltage (bottom row). Inset at bottom left shows low-magnification view; arrow represents direction of change in V and I _membrane over time. I _membrane represents the sum of all transmembrane currents and was calculated from the relationship I _membrane = I _DC − CdV/dt. Voltage threshold (V _∗) is identifiable as an inflection point in the top row and as a local peak in the bottom row. Adaptation was removed for the analysis shown here, but the shunting-induced shift in V _∗ is unaffected by inclusion of either I _M or I _AHP in the model (data not shown). B, Overlying the initial responses for the low and high conductance states highlights the change in input resistance caused by shunting (compare black lines). Responses diverge from the black lines at depolarized voltages because of activation of the sodium current. The peak is defined by the voltage at which the inward (sodium) current increases more (because of voltage-dependent activation) than the outward (leak) current increases (because of increased driving force), meaning that depolarization beyond V _∗ causes net I _membrane to decrease. The sodium current must activate more in the high conductance state than in the low conductance state (compare lengths of vertical arrows) to counterbalance the leak current and reverse the direction of change in I _membrane as voltage increases. Greater sodium current activation requires greater depolarization, explaining why an increase in membrane conductance causes an increase in V _∗. C, For experimental data, V _∗ was estimated from the voltage at which d ² V/dt ² (which reflects the first derivative or rate of activation of I _membrane) exceeded a cutoff value defined as five times the root-mean-square noise in the baseline d ² V/dt ² trace. Using responses to an arbitrarily chosen stimulus intensity of 40 pA greater than rheobase, we analyzed the second spike within each spike train because the second spike occurs before adaptation develops and its analysis is not confounded by initial membrane charging, as can occur with the first spike. Based on those measurements, introduction of a 10 nS shunt caused a significant increase in V _∗ (p < 0.01, paired t test; n = 6) that averaged 2.3 ± 0.5 mV (mean ± SEM).

Figure 7.

Spike-time precision is significantly improved by the prevention of repetitive spiking achieved through the nonlinear interaction between shunting and I _M-mediated adaptation. A, Raster plots show spike times for different combinations of adaptation and membrane conductance. On each trial, the model neuron was stimulated with a 20-ms-long pulse (stimulus trace at bottom, rectangle on raster) superimposed on the DC. Pulse magnitude was one-tenth the DC magnitude (4 and 11 μA/cm² for the low and high conductance states, respectively). For I _M-mediated adaptation in the high conductance state (bottom left), evoked spikes showed little jitter. In all other cases, background spikes and perithreshold voltage fluctuations caused evoked spikes to be less precisely timed. B, Cumulative probability distributions summarize variability in the latency of the evoked spike based on 50 trials in each condition. For I _M-mediated adaptation (left), median latencies were 4.1 and 4.2 ms for the low and high conductance states, respectively, but distributions were significantly different (p < 0.05, Kolmogorov–Smirnov test). For I _AHP-mediated adaptation (right), the median latency was 5.4 ms in the low conductance state but decreased to only 2.3 ms in the high conductance state (p < 0.001, Mann–Whitney test), but, after equalizing the median to 0 (i.e., subtracting median latency from each measurement), the two distributions were not significantly different (inset). C, After equalization (middle column), spike latency distributions were not significantly different between I _M- and I _AHP-mediated adaptation in the low or high conductance states (top and bottom rows, respectively). However, if the distributions were normalized by dividing the equalized latencies by the original median (right column), the distributions were significantly different for the high conductance state (p < 0.01, Kolmogorov–Smirnov test) but remained unaltered in the low conductance state. Equalization and normalization are used as analytical tools to distinguish between mechanisms affecting spike latency distributions; they do not imply that any comparable process is performed by the nervous system. D, In the high conductance state, if stimulus amplitude was decreased, the model neuron with I _M stopped responding (data not shown), whereas the model neuron with I _AHP kept responding but became less precise; for example, after reducing the stimulus from 10 to 3% of the DC offset, a spike was still elicited in 80% of trials, but the median latency increased significantly from 2.3 to 4.7 ms (p < 0.001, Mann–Whitney test) (left). The equalized latency distribution for 3% stimulation was significantly wider than for 10% stimulation (p < 0.01, Kolmogorov–Smirnov test) (middle), but this difference was abolished by normalization (right). Effects of equalization and normalization demonstrate that compression of the leftward skew by reduction of median latency (as with I _AHP) is not equivalent to the symmetrical narrowing of the distribution (as with I _M) and suggests that a differential sensitivity to stimulus intensity may be important for understanding how adaptation and shunting control spike-time precision.

Figure 8.

Activation of I _M at subthreshold potentials ensures high spike-time precision at low firing rates by modulating sensitivity to stimulus fluctuations. A, Raster plots show spike times for different combinations of adaptation and conductance levels. In each trial, the model neuron was stimulated with the same dynamic stimulus (shown at bottom) generated by an Ornstein–Uhlenbeck process. Stimulus amplitude was scaled by σ_signal, which was 1.8 and 4 μA/cm² for the low and high conductance state, respectively, to elicit similarly sized voltage fluctuations. For this and all subsequent panels, I _DC was 40 and 110 μA/cm² for the low and high conductance states, respectively. B, Regardless of the current responsible for adaptation, spike-time precision calculated from 45 pairwise comparisons from 10 trials in each condition (see Materials and Methods) increased as σ_signal increased. Precision was consistently less with I _AHP-mediated adaptation compared with I _M-mediated adaptation for equivalent σ_signal, but these data did not reveal an interaction between shunting and adaptation because the difference in precision between I _M- and I _AHP-mediated adaptation remained unchanged as conductance increased. SD_V, SD of voltage fluctuations. C, Firing rate, conversely, was influenced differently by σ_signal and g _shunt depending on the current responsible for adaptation. With I _AHP, firing rate was relatively insensitive to changes in σ_signal or g _shunt, whereas with I _M, firing rate was dramatically reduced by increased membrane conductance despite high σ_signal. D, To investigate this further, σ_signal was systematically varied while measuring firing rate. Shunting caused a dramatic reduction in sensitivity to stimulus fluctuations when adaptation was mediated through I _M, restricting spike generation to σ_signal > 3 μA/cm², but had virtually no effect with I _AHP-mediated adaptation. This is because I _AHP-mediated adaptation aims to maintain a constant firing rate, whereas I _M-mediated adaptation aims to maintain a constant membrane potential (see below). Inset shows relationship between σ_signal and SD_V. E, With ḡ _Na and ḡ _K set to 0 mS/cm² to prevent spike generation, comparison of average depolarization (top) reveals the outward rectification caused by activation of I _M at perithreshold potentials. Unlike the influence of I _M on average membrane potential, its slow kinetics restrict it from modulating the amplitude of fast voltage fluctuations (bottom), in which there is no difference in SD_V compared with the model neuron with I _AHP-mediated adaptation. Data are for testing in the high conductance state with σ_signal = 4 μA/cm². The resulting gap between average depolarization and voltage threshold ensures that neurons with I _M generate spikes only in response to large stimulus fluctuations when operating in a high conductance state. F, By responding exclusively to large stimulus fluctuations, the shunted neuron with I _M can maintain high spike-time precision while firing at low rates, a feature that is not evident with other combinations of adaptation and membrane conductance level. G, Activation/deactivation of g _adapt (bottom traces) after a spike (top traces) for stimulus parameters producing f _steady-state ≈ 18 Hz. I _AHP modulates interspike interval by transiently activating and deactivating, thereby enforcing a certain regularity in the spike train. In contrast, activation of I _M is relatively constant throughout the interspike interval, especially in the high conductance state, producing tonic hyperpolarization that controls responsiveness to stimulus fluctuations.