Bifurcation analysis of a two-compartment hippocampal pyramidal cell model

In the original paper, as the applied current to the somatic or dendritic compartment was increased, the authors saw various qualitative shifts in behaviour between resting, bursting and spiking (Pinsky and Rinzel 1994). Understanding the dynamic mechanisms behind these transitions is important because, within a neural network, individual Pinsky-Rinzel cells do not operate in isolation but rather there are various factors which might affect the net current impinging onto the somatic and dendritic compartments, most notably excitatory and inhibitory synaptic input and also the presence of neuromodulatory factors. Therefore we begin by investigating the effect of somatically or dendritically applied current on the dynamical behaviour of the cell using bifurcation analysis.

The bifurcation diagram using I _Sapp as the initial bifurcation parameter is formed of an S-shaped curve of steady states and a curve of periodic orbits (Fig. 3a). The lower branch of the S-curve consists of stable nodes (see Fig. 3di, I _Sapp=−1) which correspond to the hyperpolarised resting state of the cell. The system is driven to these nodes predominantly by the potassium-based I _Leak, in accordance with experimental work (Bernstein 1902; Hodgkin and Horowicz 1959). Stability is lost via a saddle-node bifurcation (SN1) at I _Sapp=0.02651, which represents the rheobase for I _Sapp and gives rise to a branch of saddles, which forms the middle and part of the upper branches of the S-curve. This branch of saddles turns around at another saddle-node bifurcation (SN2) at I _Sapp=−81.57 before regaining stability at the supercritical Hopf bifurcation (HB) for I _Sapp=23.69. For increasing I _Sapp there remains a branch of stable nodes, corresponding to a depolarised resting state of around -30mV (see Fig. 3di, I _Sapp=25).

A branch of stable periodic solutions emerges from the HB with low amplitude and high frequency (see Fig. 3dvi, I _Sapp=22). These solutions are only apparent in a small range of I _Sapp since stability is lost for decreasing I _Sapp between two torus bifurcations (TR1 at I _Sapp=21.14 and TR2 at I _Sapp=15.87). Within this range, an undulating spiking pattern is observed with increasing and then decreasing spike amplitude (see Fig. 3dv, I _Sapp=17). For I _Sapp below that at TR2, a second branch of stable periodic solutions emerges, representing regular spiking activity (see Fig. 3div, I _Sapp=3) with greater spike amplitude but lower frequency than the first branch of stable spiking solutions (Fig. 3b), as described in the original model (Pinsky and Rinzel 1994). Such spiking is dominated by the dynamics of I _Na and I _K in the somatic compartment, as well described by seminal work by Hodgkin and Huxley (1952). The aperiodic and spike doubling behaviour for lower I _Sapp (see Fig. 3diii , I _Sapp=2) is brought about via the period doubling (PD) bifurcation which occurs at I _Sapp=2.288 and leads to a final branch of unstable periodic solutions. For further decreasing I _Sapp, aperiodic and spike doubling behaviour transitions to very low frequency (VLF) bursting as in the original model (see Fig. 3dii). The currents involved in shaping these latter behaviours are discussed in more detail later in this paper. Finally, the periodic solutions disappear via a homoclinic bifurcation (HC) at I=−12.35 at which point the period is infinitely large (Fig. 3b). A homoclinic rather than saddle-node on infinite circle (SNIC) bifurcation was confirmed by showing that a very high period (1.27×10⁷) limit cycle joins together with a saddle for this value of I _Sapp, rather than coalescing with the saddle at the SN1 bifurcation (Fig. 3c).

It is important to note that for the numerical continuation results presented in Figs. 3a and 4a, we consider a very large range of the bifurcation parameter values, namely I _Sapp/Dapp∈[−500,500], in order to locate the Hopf bifurcations giving rise to stable spiking solutions. The large range of the bifurcation parameter increases the likelihood of finding turning points for the curve of steady states. Interpreting the entirety of this range as meaningful should be cautioned. It is extremely unlikely that the currents required to reach the solutions between the Torus bifurcations and between the Torus and Hopf bifurcations in the model would ever be physiologically relevant, given that the unitary amplitude of a single CA3-CA3 monosynaptic EPSC is of the order of 20–30 pA (Debanne et al. 1995). Therefore these solutions are not considered any further.

The bifurcation diagram, using I _Dapp as the bifurcation parameter (Fig. 4a), is qualitatively very similar to that for I _Sapp, although there are some differences mainly in the curve of steady states. The bottom branch of this curve consists of stable nodes, representing the hyperpolarised resting state (see Fig. 4di, I _Dapp=−1). Stability is again lost via a saddle-node bifurcation (SN1) at I _Dapp=0.02728, leading to a branch of saddles which again turns around at another saddle-node bifurcation (SN2) at I _Dapp=−83.33, before regaining stability again via a supercritical Hopf bifurcation (HB) at I _Dapp=99.78. Notably, I _Dapp at the HB is much greater than that for I _Sapp, and thus there is a larger range of I _Dapp for which there are periodic solutions, despite the rheobase being similar. The branch of stable nodes arising from the HB, which represent the depolarised resting state (see Fig. 4di, I _Dapp=120) this time loses stability at a third saddle-node bifurcation (SN3) at I _Dapp=127.6, giving rise to a final branch of saddles.

The branch of stable periodic spiking solutions emanating from the HB again grows in amplitude and period with decreasing I _Dapp (see Fig. 4dvi, I _Dapp=70) until stability is lost between the two torus bifurcations (TR1 at I _Dapp=15.59 and TR2 at I _Dapp=28.75). Within this unstable branch, there is again a form of waxing and waning spike amplitude (see Fig. 4dv, I _Dapp=25). For decreasing I _Dapp below that of TR2, a branch of stable, regular periodic spiking solutions appears (see Fig. 4div, I _Dapp=12) before stability is lost again for I _Dapp<9.127 at a period doubling bifurcation (PD). The final unstable branch of periodics for low I _Dapp disappears at I _Dapp=−3.486 again via a homoclinic bifurcation (HC). Within this unstable branch, the system transitions for decreasing I _Dapp from doublet activity for I _Dapp between ≈ 7-9, to a mixture of single spikes and doublets for I _Dapp of ≈ 6, to then aperiodic activity with a varying number of spikes per burst between I _Dapp of ≈ 2.6 and 5 (see Fig. 4diii, I _Dapp=4), to finally periodic VLF bursting between HC and I _Dapp of ≈ 2.5 (see Fig. 4dii, I _Dapp=0.3). Again the HC was confirmed by showing a very high period limit cycle solution (1.77×10⁷) for this value of I _Dapp (Fig. 4b), which we use as an approximation of the homoclinic orbit, and with which the SN1 point did not coalesce (Fig. 4c).

It is not entirely surprising that the bifurcation diagrams presented in Figs. 3a and 4a are qualitatively very similar, given that the two compartments are coupled quite closely. Nevertheless, there is clearly a larger range of I _Dapp within which the system is between the hyperpolarised resting and stable spiking states. This suggests that preferentially dendritic depolarisation could engage a wider repertoire of active behaviours and that dendritic currents could be in a more privileged position to shape these behaviours.

Indeed lowering the maximal conductance through the dendritically-located, persistent calcium channel (g _Ca) has been shown to switch bursting to spiking activity for a set value of applied current (Pinsky and Rinzel 1994; Traub et al. 1991). A change in g _Ca from 10 to 7 is sufficient to switch the behaviour of the Pinksy-Rinzel cell from intrinsic bursting to tonic firing with frequency accommodation to better represent the firing properties of CA1 cells (Taxidis et al. 2012). We therefore investigated how g _Ca affects the bifurcation structure of the system, in an attempt to understand this change mathematically.

We initially re-computed the bifurcation diagrams in I _Sapp and I _Dapp for the CA1 value of g _Ca=7 (Fig. 5). Using I _Sapp as the bifurcation parameter, the bifurcation diagram (Fig. 5a) is extremely similar to that of the CA3 cell with one noticeable difference; instead of the periodic solutions disappearing for low I _Sapp via a HC bifurcation, they disappear via a saddle-node on an invariant circle (SNIC) bifurcation. The saddle-node bifurcation of equilibrium solutions corresponding to this value of I _Sapp is that of SN1. This was confirmed by depicting in the V _s−h phase plane that the SN1 equilibrium bifurcation lies on the limit cycle at the SNIC bifurcation (Fig. 5c) and that this limit cycle used to approximate the SNIC orbit has high period (5.65×10⁵) (Fig. 5b). In addition, whereas a PD bifurcation was associated with the loss of stability of periodic solutions in the CA3 cell, no such transition was detected for the CA1 cell.

Using I _Dapp as the bifurcation parameter, there are two noticeable differences between the CA3 and CA1 cell. Firstly while the upper branch of unstable steady states did not bifurcate back to a stable solution in the range of −500≤I _Dapp≥500 for the CA3 cell, in the CA1 cell a fourth saddle-node bifurcation was detected for I _Dapp=−175.2, where the steady state solutions turn around again, giving rise to a second depolarised steady state. This means that in the range of I _Dapp between SN1 and SN3, the system is bistable (either periodic spiking and depolarised steady states between SN1 and HB, or two depolarised steady states between HB and SN3, depending on the initial parameters). It is noted, however, that a similar behaviour might be observed in the CA3 cell if the bifurcation analysis was extended beyond −500≤I _Dapp≤500 range. Secondly, no torus bifurcations were detected for the periodic spiking solutions in the CA1 cell, whereas a pair of torus bifurcations were present in the CA3 cell. Whilst this may be of mathematical interest, as discussed above, the range of I _Dapp for which this difference manifests suggests it has no physiological role. As for I _Sapp, the periodic solution disappears for low I _Dapp via a SNIC bifurcation instead of a HC bifurcation, as was the case for the CA3 cell. Therefore for all I _Sapp/ I _Dapp between the SNIC at I _Sapp=0.0556, I _Dapp=0.0574 and the HB at I _Sapp=24.01, I _Dapp=141.0 (excluding the regions between the pair of torus bifurcations for I _Sapp), the CA1 cell limit cycle solutions will be periodic spiking with decreasing amplitude and period as I _Sapp/ I _Dapp increase, with importantly no regimes of bursting or aperiodic and spike doubling behaviours.

The disappearance of these regimes is likely directly attribute to the presence of a codimension-2 bifurcation in the (I, g _Ca) parameter space (Fig. 6a-b), which causes the periodic solution to switch from vanishing via a HC to vanishing via a SNIC, as g _{C a} is decreased. The dependence of this regime change on g _{C a} also implies that C a plays an important role in shaping both irregular spiking and bursting behaviours. To examine this further, we initially explored how switches between singlet activity and doublet activity during the irregular spiking regime may be shaped by C a, for example when I _Sapp=2. As shown in Fig. 7, during the transition between repetitive singlet activity and doublet activity, the V _d and C a peaks phase advance but to different extents, leading to a reduction in the (C a−V _d) phase difference and an increase in the peak \(I_{\mathrm {K}_{\text {Ca}}}\) which drives the membrane potential to a more hyperpolarised level. This train of events to shift singlet activity to doublet activity seems plausible given changes in \(g_{\mathrm {K}_{\text {Ca}}}\), and hence \(I_{\mathrm {K}_{\text {Ca}}}\), have previously been shown to shift a cell’s behaviour from spiking to bursting (Tsaneva-Atanasova et al. 2007; Nowacki et al. 2010; Szalai et al. 2011; Iosub et al. 2015).

Finally we study how C a and the C a-dependent q dynamics shape the bursting behaviour in the model. In the original paper, the VLF bursting dynamics of the CA3 cell are described to be largely determined by q which, when passes below a threshold, was thought to trigger the initial somatic spike of the burst. As such, bursting was not thought to persist when q was replaced by its mean value (Pinsky and Rinzel 1994). Since then, the idea that q acts as the threshold for burst initiation has been called into question from phase-plane analysis on a piecewise reduced model (Booth and Bose 2001; Bose and Booth 2005). We therefore performed fast-slow analysis to probe the mathematical mechanism behind the bursting dynamics further. Since both C a and q decay on slower timescales (13.33ms and 100−1000ms respectively) than the other gating variables (<6ms within the effective ranges of V _s and V _d see Fig. 8a) (Pinsky and Rinzel 1994)) and therefore the ratio of timescales between fast and slow variables is <<1, this analysis is appropriate to investigate how both the C a and q dynamics influence bursting. The fast-slow method was originally pioneered by Rinzel (1987), and involves separating the full system into a fast subsystem and a slow subsystem, where the slow variables (in this case C a and q) can be used as parameters for bifurcation analysis of the fast subsystem. We analysed the VLF bursting dynamics for both applied somatic current and applied dendritic current. I _Sapp or I _Dapp were set to 0.3 with the other equal to 0 (values that generates VLF bursting in the full system). The same qualitative bifurcation structure was observed in both cases. Therefore we present only the bifurcation diagrams for I _Sapp, whilst describing both bifurcation values for the fast subsystem, where I _Sapp=0.3/ I _Dapp=0.3.

We begin our fast-slow analysis using q as the bifurcation parameter. For steady applied current, the bifurcation diagram of the fast subsystem (Fig. 8b) is formed of a curve of steady states and a curve of bursting periodic orbits. Since the system continues to burst while q is fixed, the intraburst dynamics are independent of q, and this variable (q) cannot act as a threshold for burst initiation, confirming results in (Booth and Bose 2001; Bose and Booth 2005). To elaborate, the curve of steady states has a lower branch of stable nodes, representing the hyperpolarised resting state, which loses stability for decreasing q via a saddle node on invariant circle (SNIC) bifurcation at q=0.1136/ q=0.1119 (see Fig. 8d) and gives rise to an upper branch of saddles. From the SNIC, a stable branch of bursting periodic solutions also emerges with initially high period (Fig. 8f). As q decreases, the amplitude of the burst remains fairly constant while the period decreases. Stability is lost at q=−0.05923/ q=−0.06627 via a period doubling bifurcation (PD1) then regained briefly via a second period doubling bifurcation (PD2 at q=−0.4779/ q=−0.4844) before being lost again via a saddle node of periodics (SNP) bifurcation at q=−0.5024/ q=−0.528. At this point, the bursting periodic orbit branch turns around and increases in period for increasing q (Fig. 8f) until it terminates via a homoclinic bifurcation (HC at q=0.2524/ q=0.2653, see Fig. 8e). Following the burst trajectory (shown in cyan in Fig. 8b), the phase point tightly follows the branch of stable nodes for decreasing q and then jumps to the maxima of the bursting trajectory once past the SNIC. In the full system, for this value of applied current, this transition takes several hundred ms (Fig. 8c), again confirming that the threshold for burst initiation is not described by the q dynamics. Midway through the burst, q then begins to increase which pushes the system back through the SNIC. After this point, the phase point transiently oscillates before returning to the branch of stable nodes. Therefore since the transition below the SNIC in q permits the occurrence of the burst, the time between q increasing above the SNIC and then decreasing below the SNIC strongly controls at least part of the interburst interval; a finding consistent with previous work (Bose and Booth 2005).

Given that q could not fully describe the intraburst dynamics, we then turned our attention to the second slow variable, C a and performed fast-slow analysis using C a as the bifurcation parameter. For steady applied current, the bifurcation diagram in C a (Fig. 9a) has two long branches of stable nodes, an upper branch representing a depolarised steady state for values of C a<127.5/ C a<127.6 and a lower branch representing a hyperpolarised steady state for C a>4.263/ C a>4.117. Stability in the lower branch is lost for decreasing C a via a saddle node bifurcation (SN1) at C a=4.263/ C a=4.117. The resulting unstable branch of saddles turns around and continues for larger C a values. Stability in the upper branch is also lost via a saddle node bifurcation (SN2) at C a=127.5/ C a=127.6. The branch turns around briefly and is formed of a series of saddles until stability is regained momentarily between another couple of saddle node bifurcations (SN3 at C a=112.5/ C a=112.6 and SN4 at C a=127.2/ C a=127.4). After SN4, the upper branch consists of a second series of saddles which turn around at a fifth saddle-node bifurcation (SN5 at C a=62.76/ C a=63.73) before continuing on to larger values of C a. Between SN4 and SN5, there is a subcritical Hopf bifurcation (HB) at C a=112.7/ C a=113.9 which gives rise to the unstable periodic solutions. For decreasing C a, the amplitude and period of the solution increases (Fig. 9c), before turning around at the saddle node of periodics (SNP at C a=11.21/ C a=11.92) and terminating via a homoclinic bifurcation (HC at C a=14.58/ C a=15.23) (Fig. 9d). Following the burst trajectory, it can be seen that the phase point, loosely follows the lower branch of stable nodes for decreasing C a and overshoots SN5 before jumping to the upper branch of stable nodes. Again the attraction is weak, because the C a dynamics are only intermediately slow, and so the phase point transiently oscillates around this upper branch. Even when the phase point is past the HB, there are smaller oscillations around the unstable upper branch before returning to the attraction of the lower branch of steady states. VLF bursting is therefore governed by the C a dynamics and is of the fold/subHopf type, adoping the nomenclature in Izhikevich (2000), which has been reported also for biophysical models of pituitary somatotrophs and lactotrophs (Stern et al. 2008; Tsaneva-Atanasova et al. 2007; Tabak et al. 2007).

The discontinuous functions in the original Pinsky-Rinzel model were modified to facilitate numerical bifurcation analysis on the full system (Fig. 1), whilst maintaining the qualitative behaviour of the original model (Fig. 2). We show that spiking behaviour emerges in the model from a supercritical Hopf bifurcation and that the rheobase is formed from a saddle-node bifurcation (Figs. 3 and 4). This mechanism of spiking is therefore similar to the bifurcation structure, described earlier, for a single compartmental CA3 model (Grobler et al. 1998). As g _Ca is decreased, a codimension-two SNIC bifurcation switches the periodic spiking solutions from terminating via a homoclinic bifurcation to terminating via a SNIC bifurcation (Fig. 6). This eliminates the unstable periodic regime, where bursting and irregular spiking solutions exist, and thus explains why a reduction in g _Ca leads to unperturbed spiking for increasing I _Dapp and a larger parameter range of I _Sapp (Fig. 5), as previously observed (Pinsky and Rinzel 1994; Traub et al. 1991). The disappearance of irregular spiking as g _Ca decreases, is suggestive that C a, as well as shaping bursting behaviour, is also important in shaping spiking in the model. Indeed we show that a reduction in the (C a- V _d) phase difference, and corresponding increase in \(I_{\mathrm {K}_{\text {Ca}}}\) is likely to drive activity from singlet spiking to doublet/triplet spiking in this irregular spiking regime (Fig. 7).

As an example of how understanding where Pinsky-Rinzel cells sit dynamically is important for understanding spiking in larger networks, we turn out attention to a combined CA3 and CA1 hippocampal network model, of Pinsky-Rinzel cells that exhibits sharp wave ripple (SWR) oscillations (Taxidis et al. 2012). SWRs are transient network events that occur during periods of awake immobility and slow wave sleep (Buzsaki et al. 1983; Buzsaki 1986; O’Neill et al. 2010), and are known to be important for spatial learning and memory (Girardeau et al. 2009; Ego-Stengel and Wilson 2010; Jadhav et al. 2012). During sharp wave ripples, hippocampal place cells that fire in discrete locations of an environment (O’Keefe and Dostrovsky 1971), reactivate such that co-active place cell activity, representing a discrete environmental location, or sequences of place cell activity, representing a trajectory through an environment, are replayed (Wilson and McNaughton 1994; Nadasdy et al. 1999; Lee and Wilson 2002; Foster and Wilson 2006; Csicsvari et al. 2007; Diba and Buzsaki 2007; Davidson et al. 2009; Karlsson and Frank 2009). However the mechanisms that select which place cells are active in SWRs is still unclear, although the intrinsic excitability of the cells and the local synaptic connectivity structure of the network within which they reside are both likely to play a role (for review see Atherton et al. (2015)).

In the Taxidis model (Taxidis et al. 2012), CA1 cells are driven to spike by CA3 cell activity. From a dynamical systems perspective, the occurrence of a single CA1 spike could be influenced by CA3 in two ways. Firstly, if the CA1 cell was in a spiking limit cycle regime, excitatory input from CA3 could phase shift CA1 cell spike times. Secondly, if the CA1 cell was in a hyperpolarised resting state, excitatory input from CA3 could push the CA1 cell trajectory through the phase space in a manner that resembles a spike, but which does not actually push the CA1 cell trajectory onto a stable limit cycle i.e. causes a transient. The bifurcation analysis, conducted here, facilitates the differentiation between these quite different mechanisms of CA1 spiking, namely periodic attractor driven verses transient oscillations. The analysis shows that for the parameter values of I _Sapp and g _Ca used in the model (0 and 7 respectively), the CA1 cells are in a hyperpolarised resting regime below the SNIC. Therefore at least in this model, it can be concluded that CA1 cell spiking in SWRs is driven by transient oscillations, induced by presynaptic CA3 cell activity, and not by a periodic attractor driven excitability.

Experimental findings have shown that the intrinsic bursting of CA3 cells is driven by several processes: the intital fast spike is driven by activity through sodium channels; the ensuing after depolarising potential (ADP) subsequently drives the burst; and the burst is then terminated by an after hyperpolarising potential (AHP) (Wong and Prince 1978; 1981). Although the ionic bases for the ADP and AHP that occur during bursting in CA3 pyramidal cells are not fully understood, they are both calcium driven (Wong and Prince 1978; 1981; Schwartzkroin and Stafstrom 1980) and are likely mediated by T/R-type calcium channels and calcium-activated potassium channels respectively, as has been found in some cases for CA1 cell bursting (Magee and Carruth 1999; Metz et al. 2005; Alger and Nicoll 1980), although see (Azouz et al. 1996). Kv7/KCNQ/M-channels also play a role in terminating the ADP (Vervaeke et al. 2006; Brown and Randall 2009).

Previous studies using fast-slow analysis of CA3 pyramidal cell models have proposed different mechanisms as underlying bursting activity (Kepecs and Wang 2000; Xu and Clancy 2008). In a reduced version of the Pinsky-Rinzel model with similar somatic currents but only a persistent sodium current and a slow potassium current in the dendritic compartment; bursting activity was found to be of the square-wave/fold-homoclinic type and dependent on the activation variable of the slow potassium current (Kepecs and Wang 2000). In contrast, using fast-slow analysis on a single-compartmental CA3 model, incorporating various additional currents to the Pinsky-Rinzel model, bursting was found to be of the parabolic type and dependent on two variables: the slow autocatalytic activation variable for a T-type calcium channel and the activation variable for the slow calcium-dependent potassium current (Xu and Clancy 2008).

It remained to be established which, if either, of these mechanisms is true for the original Pinsky-Rinzel model, where the two candidate slow variables for controlling bursting are calcium and the activation gate, q, of the dendritic after hyperpolarisation current. Here, we identify a mechanism intermediate between the two cases, whereby bursting is of the fold/sub-Hopf type and relies on just one variable, calcium, which is dependent on the dendritic calcium current, I _Ca, and in fact controls both the calcium-dependent potassium current, \(I_{\mathrm {K}_{\text {Ca}}}\), and the activation gate, q, of the dendritic after hyperpolarisation current, \(I_{\mathrm {K}_{\text {AHP}}}\). The rise of C a through the subcritical Hopf bifurcation mediates the slow termination of the burst (Fig. 9), thereby biophysically explaining previous results on a piecewise reduced system, showing disruption of the ping-pong mechanism and termination of the burst as I _Ca was activated (Booth and Bose 2001; Bose and Booth 2005). The results, here, are also consistent with the blockade of calcium channels perturbing CA3 bursting Wong and Prince (1978, 1981) which could not be explained by bursting dependent on activation of a slow potassium current alone (Kepecs and Wang 2000).

The original Pinsky-Rinzel paper proposed that q sets the burst initiation threshold (Pinsky and Rinzel 1994). Subsequent work, using the piecewise reduced system, showed this threshold was independent of q and instead set by V _d (Booth and Bose 2001; Bose and Booth 2005). We clarify this point here by showing, using fast-slow analysis, that while q plays no role in setting the threshold for the initial somatic spike in the burst or indeed the intraburst dynamics, it does set part of the interburst interval (Fig. 8). The biophysical implications of this is that the dendritic afterhyperpolarisation current plays an important role in setting the interburst interval.

Nevertheless, it it important to note that the dependence of bursting on just one slow variable is likely due to the bistability of the fast subsystem in the Pinsky-Rinzel model which is driven by the window I _Na (Golomb et al. 2006). This derives from the overlap in the steady-state activation and inactivation curves for the I _Na current, where the inactivation curve is shifted to more depolarised values (>20m V) compared to experimental results (Sah et al. 1988). When this window current was reduced, bursting in a different CA3 model was instead dependent on two slow variables and was of the parabolic type (Xu and Clancy 2008).