Control of sleep-to-wake transitions via fast aminoacid and slow neuropeptide transmission

Abstract

The Locus Coeruleus (LC) modulates cortical, subcortical, cerebellar, brainstem and spinal cord circuits and it expresses receptors for neuromodulators that operate in a time scale of several seconds. Evidences from anatomical, electrophysiological and optogenetic experiments have shown that LC neurons receive input from a group of neurons called Hypocretins (HCRTs) that release a neuropeptide called hypocretin. It is less known how these two groups of neurons can be coregulated using GABAergic neurons. Since the time scales of GABA_A inhibition is several orders of magnitude faster than the hypocretin neuropeptide effect, we investigate the limits of circuit activity regulation using a realistic model of neurons. Our investigation shows that GABA_A inhibition is insufficient to control the activity levels of the LCs. Despite slower forms of GABA_A can in principle work, there is not much plausibility due to the low probability of the presence of slow GABA_A and lack of robust stability at the maximum firing frequencies. The best possible control mechanism predicted by our modeling analysis is the presence of inhibitory neuropeptides that exert effects in a similar time scale as the hypocretin/orexin. Although the nature of these inhibitory neuropeptides has not been identified yet, it provides the most efficient mechanism in the modeling analysis. Finally, we present a reduced mean-field model that perfectly captures the dynamics and the phenomena generated by this circuit. This investigation shows that brain communication involving multiple time scales can be better controlled by employing orthogonal mechanisms of neural transmission to decrease interference between cognitive processes and hypothalamic functions.

1. Introduction

Cognitive processes like speech or object recognition are carried out very quickly in the brain. Within 300 milliseconds of neural processing we can already recognize objects [1–3]. Since there are many neuron and circuit layers, this information is possibly transmitted and regulated by the fastest networks of synaptic connections made of rapid aminoacids like AMPA [4] and GABA_A receptors [5].

An example of a harder computational problem is speech articulation [6]. It operates at larger time scales because it coordinates motor control and requires the conversion of thoughts into an ordered list of messages. The synaptic receptors that can control these 100–200 ms time scales can be NMDA receptors [7] and GABA_B receptors that can extend their inhibitory effect as long as half a second. The decay times of NMDA receptors can extend beyond 200 milliseconds and GABA_B receptors strongly depend on the firing rates of the inhibitory neurons. This activity-dependent variability of GABA_B provides a rich repertoire of time scales and neural codes [8]. These cognitive processes appear to have all the necessary processing tools to operate in the sub second time scale.

However, the brain is not only computing fast cognitive processes. It needs to rest, sleep, feed, and repair. Lack of sleep, for instance, results in a significant impairment of cognitive tasks interfering with other sensory-motor activities and memory formation [9]. The brain needs an operating system to provide the conditions such that all the brain circuits cooperate with little disturbance to each others function. Sleep cycles are a key aspect of the brain operating system. This key function extends well above the cognitive time scales using minutes and hours. The mechanisms of neuro transmission are then not solely relying on fast aminoacid communication, but a new set of neural tools are needed in the form of neuropeptides to depolarize or inhibit brain circuits [10, 11].

While aminoacid transmission operate in the nanometer spatial scale, neuropeptides may be released extrasynaptically to cover micrometers. This makes neuropeptide transmission less selective, as it sends signals not neuron-to-neuron exclusively but to all neurons in a region with a specific type of neuropeptide receptor.

Among the many neuromodulators involved in the function with these longer time scales, the Hypocretin neuropeptides, produced by a few thousand neurons in the lateral hypothalamus (referred to as HCRT neurons herein), stand out as critical regulators of sleep/wake cycles [12, 13]. Recently, optogenetic studies have shown how the Locus Coeruleus (LC) [14, 15], a brain structure localized in the brainstem, mediates the sleep-to-wake transition induced by hypocretin/orexin neuropeptides [16]. The HCRT population, which projects into the LC cells [17–20], presents bursts of activity preceding the wake transition, exciting the LC cells which in turn induce sleep-to-wake transition [21–24]. This induces a transition of physiological state of the whole organism [25].

The output of LC neurons is likely to be regulated by GABAergic cells in the sublaterodorsal peri-LC, providing a substantial input to LC cells [26, 27]. This introduces a very intriguing interplay between two very discrepant time scales involving HCRT neuropeptides with a decay time of about minutes and GABA_A receptors in the millisecond range. The interplay of these time scales accommodating several orders of magnitude are not infrequent in the hypothalamus [28]. These deep neural circuits have been associated with the control of REM sleep atonia [29, 30] and bursts of LC overexcitation leads to (reversible) behavioral scenarios associated with neuropsychiatric disorders [21]. Therefore, imbalances in excitation/inhibition in the LC may underlie sleep disorders, posing the LC regulation as an important question to be understood in depth. Other forms of corticothalamic feedback control have been studied before as a key gear underneath brain oscillations (e.g., spindles) during slow-wave sleep [31–35].

Here we investigate through realistic and reduced mean field models the interplay between a fast aminoacid (GABA) and a slow neuropeptide (HCRT) in the sleep-to-wake transition regulation. We have modeled the system as three groups: LC, HCRT and GABA populations. The main hypothesis is that the GABA inhibition controls LC activity. LC and HCRT model cells were used from previous publications [11, 16], and the GABA model is introduced in section 2. We then show that, unexpectedly, the presence of a GABA population actually increases the activity of LC cells. We use Phase-Resetting curves to understand that this is actually due to the active integration inherently present in LC cells. In order to suggest a possible regulation mechanism, we propose a new group which releases a neuropeptide acting on the same time scale of the HCRT peptide, but with an inhibitory effect on LC cells. This new population can balance the excitation that comes from HCRT cells and accurately regulates the activity of LC cells, not only decreasing its firing rate but also allowing a nimble rise up of activity on the onset of HCRT excitation as well as a swift decay at the end of excitation. We also show a very simple mean-field model, based on simplified Wilfon-Cowan’s equations, that captures the dynamics of this circuits. We then discuss how our hypothesis can be experimentally tested.

2. The network model

In previous studies [11, 16], we have used two conductance-based compartment models to describe LC and HCRT cells, as well as chemical rate equations for the release of neurotransmitters to simulate AMPA synapses. These models were fitted to agree with electrophysiological recordings of neural populations present in the hypothalamus and are described elsewhere [16]. Here we add a third group, probably located at the peri-LC, which inhibits both HCRT and LC cells through GABA_A receptors (see Fig. 1). For the j-th GABA cell, we use two compartments: the soma,

Figure 1.

Network connectivity. There are three neural population. Excitatory connections from HCRT cells are made through AMPA and the release of hypocretin peptides (HCRT). LC cells can excite other cells in the same group via AMPA neurotransmitter. Inhibitory synapses are based on the release of GABA amino acid and the presence of GABA_A receptors. The gray numbers next to each connection are the connection probabilities among the groups. Unless explicitly stated, we use 20 neurons per population.

$$C_S\frac{{dV}_j^S}{dt}(t)=-g_{SL}\left(V_j^S(t)-V_L\right)-I_A-I_{Ca}-g_{AS}\left(V_j^S(t)-V_j^A(t)\right),$$ (1)

and the axon,

$$C_A\frac{{dV}_j^A}{dt}(t)=-g_{AL}\left(V_j^A(t)-V_L\right)-I_{Na}-I_{Kd}-I_{K[Ca]}-g_{AS}\left(V_j^A(t)-V_j^S(t)\right).$$ (2)

Note that we have not included any low threshold currents, even though they are present in both LC and HCRT cell models. We have used g_SL = 0.96nS, g_AL = 0.48nS, V_L = −61.6mV, C_S = 10pF, C_A = 5.0pF and g_AS = 65nS. All remaining currents in equations 1 and 2 are described in Appendix A.

We model the GABAergic synapses similarly to the AMPA and HCRT connections previously used [16, 36]. The release of GABA neurotransmitter by neuron j, which should bound at GABA_A receptors in LC and HCRT cells, evolve according to the kinetic equation

$$\frac{{dr}_j}{dt}(t)=k_{\mathit{GABA}}^f(1-r_j(t))\mathrm{\Gamma }\left(V_j^S(t),30\text{mV},-2\text{mV}\right)-k_{\mathit{GABA}}^r{r}_j(t),$$ (3)

with the rise constant $k_{\mathit{GABA}}^f=100{\text{ms}}^{-1}$, decay constant $k_{\mathit{GABA}}^r=0.7{\text{ms}}^{-1}$ and $\mathrm{\Gamma }(x,y,z)=1/\left(1-e^{\frac{x-y}{z}}\right)$. Then the current into a cell labeled k, in either the LC or HCRT populations, is defined as

$$I_k^{\mathit{GABA}}(t)=\sum _{j=1}^{N_{\mathit{GABA}}}{g}_{jk}^{\mathit{GABA}}{r}_j(t)\left(V_j^S(t)+90\text{mV}\right),$$ (4)

with N_GABA being the number of GABA neurons and $g_{jk}^{\mathit{GABA}}$ representing the synaptic strength between neurons j and k, measured in nS. For simplicity, if j and k are connected, then $g_{jk}^{\mathit{GABA}}=g^{\mathit{GABA}}$; else, $g_{jk}^{\mathit{GABA}}=0$. To draw the connections, we set a probability of connection between groups and then choose pairs of neurons uniformly. Following observations of the existence of indirect GABAergic inhibition from the HCRT population [37], the GABA neurons receive excitatory inputs from HCRT and inhibit the LC neurons (without any feedback). The connection probabilities are shown in Fig. 1.

In the simulations we used Backward Differentiation Formula and Newton iteration implemented within CVODE, with relative and absolute tolerances of 10⁻⁹ [38]. We used 20 neurons per group, unless stated otherwise, resulting in a total of 845 coupled differential equations.

Carter and collaborators [21] have shown that the stimulation of LC cells during an interval of order of some seconds results in the wake transition. Thus we have set a stimulation protocol: we inject into the HCRT neurons (soma) a current of magnitude I_DC within a 10 second time interval. Then we can assess how the LC responds to a sudden increase of activity in the HCRT cells, which is observed in experiments whenever a sleep-to-wake transition takes place. In Fig. 2 we show the activity generated by this protocol. As the HCRT neurons are excited (starting precisely at 70s), we see an immediate increase in activity of LC and GABA neurons. As the stimulation stops acting, the LC neurons eventually stop firing, which is expected. We also show some activities of individual neurons from each of the three groups, where a considerable change is seen when this external current takes effect.

Figure 2.

Detailed neural activity resulting from 10 second stimulation of HCRT cells with a 20pA current. Connectivity is shown in Fig. 1. We have set g^GABA = 100nS, which generates a weak, but significant inhibitory postsynaptic potentials (approximately 1mV). Left: average soma voltage of LC and HCRT cells. We also show the firing rate of all populations. Right: individual activity of 2 randomly chosen neurons for each population.

3. Role of the GABA population

It is likely that this GABA population regulates the LC activity, especially because an activity overload may lead to behavioral attacks similar to episodes of neuropsychiatric disorders [21]. For this reason we wanted understand exactly how the GABA population is affecting the LC (and HCRT) populations. We show in Fig. 3 [a] completely unexpected result: as the GABAergic synapses get stronger, the firing rate of LC cells increases instead of decreasing. We show HCRT and GABA inputs to a LC cell, which have completely different time scales, in Fig. 3[b].

Figure 3.

The effect of the GABA_A inhibition on the activity of the LC neurons leads to unexpected firing rate increase with the synaptic strength g^GABA enhancement. [a] Firing rate of all populations as g^GABA is raised. The particular case without the activity of GABAergic neurons is precisely the same condition considered in [11]. [b] Activity and input current of a randomly chosen LC neuron. The time scales of HCRT and GABA are completely different. During HCRT stimulation, GABA current peaks have a smaller amplitude due to the relative refractory time (originated by the delayed K⁺ current). Nevertheless, it is clear that the frequency of these pulses increases significantly, driven by the HCRT estimulation. [c] We show that if the current that excited HCRT cells remains for enough time, then the firing rates of all three populations achieve an equilibrium (on the left a simulation with g^GABA = 500 nS is shown). Then we use these steady firing rates to show (right panel) the decrease of LC firing rate as g^GABA grows.

In Fig. 3[c] we show that, by indefinitely exciting the HCRTs, the firing rates of all three populations eventually reach a steady level, which mainly depends on g^GABA. The steady firing rates may also depend on other parameters (synaptic strength between HCRT and LC groups, amplitude of the HCRT excitation, connectivity probabilities, etc), but we focus on its dependence on g^GABA to investigate the control of LC activity. Then, averaging these steady firing rates over time, we can see the LC firing rate dependence on g^GABA (see Fig. 3[c], right panel). It is clear that both GABA and LC neurons increase their activities, showing that this effect occurs in a wide range of g^GABA. Since HCRT receive feedback connections from GABA (see Fig. 1), we see a meager increase in the firing rate (≈ 1.0Hz) of the GABA neurons as g^GABA becomes stronger due to a mildly synchronization of HCRT neurons.

To help make sense of g^GABA values we present in Fig. 3, we have checked, by direct inspection of the postsynaptic membrane potential, if the amplitude of the Inhibitory Postsynaptic Potential (IPSP) happened to be in an acceptable range (visible in Fig. 4[b]). Throughout all g^GABA values considered, the IPSP amplitudes are always lesser than 4mV, which may be considered a reasonably strong inhibition in most systems.

Figure 4.

How LC neurons, whose parameters were fitted to behave as in previous experiments [16], behave when receiving GABA neurotransmitter. [a] Two cells, one LC and one GABA, are simulated. First we insert a weak current into the LC soma to make it spike intermittently. After a while (20s), the GABA neuron receives a similar current and starts firing, inhibiting the LC cell. Although pulses of membrane hyperpolarization are observed in the LC voltage time series, the firing rate of the LC cell actually increases significantly. [b] Membrane potential of both perturbed and unperturbed (reference) model LC neurons. Although an inhibitory pulse is used – which generates the hyperpolarization around 9s –, the next spikes get advanced with respect to the unperturbed case. [c] Phase-Resetting curves (PRCs) of the LC cells for an inhibitory bi-exponential pulse, fitted from the output GABA currents, with several intensities I_p. Notice that prompting such inhibitory perturbation in a considerably range of phases (for most of cases, over 80% of the whole range) actually results in an advance of phase (same effect seen in [b]). This is the root cause for the increase in firing rate.

This result is puzzling and opens space for a few questions. For instance, if we simulate the exact same system, but with a GABA population slightly larger, will the LC firing rate get smaller? We have tested it with a GABA population consisted of 30 and 40 neurons (i.e., twice bigger than the LC and HCRT populations) and the result is that the LC firing rate can get even a little higher. Also, in general spikes induced by inhibition are usually due to low-threshold currents, and LC neurons present both calcium I_T and I_h low-threshold currents. By shutting both currents down (independently), this increase in firing rate with g^GABA persists, thus it is not caused by the presence of such currents. Finally, this result does not substantially change if we modify the connection probability from LC group onto itself. The only restriction is that this probability must be higher than ≈ 0.2, below which exciting the LC group becomes much slower and does not comply with experimental observations. To some extent, these loopback connections help the LC group to engage some activity level quickly.

One last question is whether this whole effect is robust to changes in the decay time constant $1/k_{\mathit{GABA}}^r$, defined in Section 2. This time constant actually defines the time scale of the GABA_A action on LC cells, and reports show that this constant can be as large as 8 ms. In fact, there are observation of two kinds of GABA_A inhibitory postsynaptic currents [39]: GABA_A,fast, which accounts for this range of decay constant, and GABA_A,slow, which manifest a decay constant larger than 30ms and is present in all brain, though less frequent. We started investigating the GABA_A,fast using several values in the range 1.5–8 ms and the same effect persists. Also, for a time constant of about 5–6 ms, the increase in firing rate becomes even more pronounced.

However, when we have used a decay time scale consistent with GABA_A,slow (namely, 30–100ms), the GABA population was able to decrease the firing rate of LC cells. Although at first sight this might be a possible mechanism for LC activity regulation, there are reasons that suggest otherwise, such as the fact that most connections into LC neurons are gap junctions [40], ruling out the GABA_A,slow known underneath mechanisms (diffusion and affinity of GABA_A receptors) [39]. This makes this hypothesis very unlikely, but still possible. Also the observed inhibition in our simulations is extremely noisy, introducing not observed oscillations in the LC activity (not shown), and thus it does not features as a good controll mechanism as well as the hypothesis we shall present in section 5.

4. Inhibition-induced advance in spike timing

GABA is the main inhibitory neurotransmitter present in mammalian central nervous systems, yet it is increasing the firing rate of LC neurons somehow. In Fig. 4[a], we isolate this effect by simulating only two neurons: one LC cell and one GABA cell. The GABA connects to LC via equation (4) and they both receive a steady input current, starting at different times. First a 26pA current is injected at the LC cell, making it fire intermittently at a constant rate of nearly 2.7 Hz (comparable to the behavior of these cells during awake states [21]); then, after 20 seconds, the GABA neuron receives a similar current and starts perturbing the LC neuron. One can see that the firing rate of the LC gets twice larger at this moment, even though the voltage time series of the LC shows IPSPs (see 4[b], more details below). To maximize this effect, we have used g^GABA = 3 × 10³nS. We next use Phase-Resetting curves [41–43] to understand what happens when a LC cell receives a inhibitory presynaptic potential.

Phase-Resetting curves (PRCs) are widely used to investigate synchronization and phase-locking issues, both theoretically [44] and experimentally [45, 46]. The main idea is to asses, once an oscillator is trapped on a limit cycle, how does it respond to an external stimulus prompted at different phases ϕ of its cycle. If after receiving this stimulus the oscillator remains in the basin of attraction of its limit cycle, then eventually it will return the original oscillation, but with a different phase, given the trajectory deviation. A PRC is the graph of this phase difference Δϕ against the phase ϕ at which the stimulus is prompted to the oscillator. Following our definition of PRC [41], a positive Δϕ means advance in spike timing – the next spike happens sooner; a negative Δϕ means a delay.

In Fig. 4[b] we show our protocol to assess the PRCs. We injected a current into LC cells to make it spike intermittently, with a steady frequency of approximately 3 spikes per second. The inhibitory pulse is prompted near 9.0s. We fitted the GABA_A receptor currents from our model and used it as stimulation, with a scaling factor I_p to make the pulse as strong as one wants. There is a direct correspondence between g^GABA and I_p, but they’re not exactly the same, since I_P is just a factor in the fitted bi-exponential formula^§. Then, after a small transient (usually taking no more than one spike), the phase difference Δϕ is measured. This leads us to the PRCs we show in Fig. 4[c], which are characteristics of the so-called Type II oscillators (having positive and negative Δϕ). In the majority of cases, more than 80% of the phase range has a positive Δϕ, which means that, although the inhibitory pulse causes a hyperpolarization in the cell’s membrane voltage, the ending result is an advance in the forthcoming spikes.

Also, in Fig. 4[b] we can already see this phase advance effect, as after an IPSP the spikes of the perturbed time series arrive sooner than the reference time series spikes. And as this phase advance is often manifested in these cells, an inhibitory train of pulses may increase the firing rate, explaining Figs. 4[a] and 3. We believe this is the effect that is driving our observations in section 3.

5. Regulation of LC activity: a hypothesis

LC activity cannot be disrupted into long bursts of activity, as it leads to sleep disorders, and it is believed that some neural population should exist to deter this undesired scenario. Through GABA neurotransmitter alone this is not accomplished. We have tested, for instance, what happens if we change the synaptic strength of HCRT group onto LC if either there is or there is not a GABA population present. In all cases tested, whenever the GABA population was present, the LC firing rate was boosted, and we see no regulation at all.

This follows from the inner characteristics of our LC neurons: they are actively integrating their inputs, with a fast rise up after a hyperpolarization. Other neural dynamics might actually showcase a decrease in firing rate with intermittent GABA excitation, especially passive models (such as leaky-integrate-and-fire models). Unfortunately, this is not the case with LC neurons, as shown in recordings in previous experiments [11]. In short, the PRC should present a considerable portion of phases with negative Δϕ, instead of positive. Unless an exotic dynamical system capable of reproducing both observed population features of LC recordings and also show an adequate PRC, GABA neurotransmitter is not the answer for LC activity regulation.

We then have formulated the following hypothesis: what if the regulation is given by the competition between two neuropeptides with large time scales (comparable to HCRT), one excitatory and another inhibitory? This would generate a balanced input current in the time scale of tens of seconds, regulating the activity of LC cells. This may not be common, but in specific cases localized peptides can exert opposite effects on postsynaptic neurons, and one example is the HCRT population itself, which receives both hypocretin as excitatory and dynorphin as inhibitory neuromodulators [47]. There are plenty of neuropeptides with synaptic currents very similar to the HCRT model we use here (see Fig. 3[b]) [48].

To answer this question, we add another population into our model which inhibits the LC population using an inhibitory neuropeptide, with time scales comparable to HCRT. We shall refer to this population as Inhibitory Neuropeptide (INP for short). For simplicity purposes, we shall use for the INP the exact same model of HCRT cells, but with inverted sign of synaptic strength. This way the net contribution to a LC neuron becomes a competition between HCRT excitation and this unknown peptide inhibition (shown in Fig. 5[c]).

Figure 5.

The addition of an inhibitory neuropeptide population inhibiting LC activity. The excitation protocol remains the same. [a] Firing rate of all populations when the inhibition from the inhibitory neuropeptide into LC cells is present. The signal becomes smaller, proportioning a possible regulation mechanism. Compare to Fig. 3[a]. [b] The membrane potential of one randomly chosen cell from each population. [c] Synaptic inputs of a single LC cell. It is possible to see how INP and HCRT balance each other and deliver a right amount of excitation to LCs. [d] How the firing rate depends on the synaptic strength g of this INP neuropeptide, using the same protocol proposed in Fig. 3[c]. The decrease in the INP firing rate is due to a lower excitation coming from LC cells.

Since the idea is that this neuropeptide acts whenever the LC population becomes overloaded with activity, a connection from LC onto this inhibitory neuropeptide makes sense. We have also connected into this population the HCRT, so that it can engage in some residual activity and respond to LCs faster. We show our main result in Fig 5[a]. This inhibitory neuropeptide is actually capable of regulating LC activity.

We also show for completeness in Fig. 5[b] the activity of a candidate neuron per group, showing a significantly increase in activity right after the LC cell starts firing. We can finally compare in Fig 5[d] how the firing rate of each population depends on the inhibitory synaptic strength g of INP neuropeptide (as opposed to Fig. 3[c]). We use the same protocol proposed before: we stimulate the HCRT population indefinitely and wait all populations to reach their stationary firing rates. With this INP population the LC firing rate decreases with g, thus suggesting that this can be a mechanism underlying the LC activity regulation. The scale of g (kS in Fig. 5[d]) matches the synaptic strength used from HCRT to LC. There are no significant differences, from this point of view, between having or not the GABA neurons, although they may be important from other aspects. The decrease in the INP firing rate is due to a lower excitation coming from the LC cells, and thus a lower recruitment.

It is important to notice that the INP population, with a right choice of parameters – without a need of a fine tuning –, combine a lower firing rate with a swift rise up of LC activity and a faster end of activity: comparing Figs. 5[b] and 3[a] we see that the activity of the LC cells dies out almost 10s before and the highest firing rate achieved during stimulation of HCRT cells is only slightly smaller. Although the onset of LC activity is delayed with the INP population (by less than 3s), the rising timescale is way faster. These features pose the INP as a nimble regulation mechanism of LC activity. This observation may be the most important result and the most compelling signal that the existence of such neurotransmitter could be of great importance to this hypothalamic circuit.

Also, this activity regulation does not only hold when the inhibitory and excitatory neuropeptides have precisely the same parameters, which would be a fairly unrealistic situation. We have found that the key feature underneath this effect is to use an inhibitory neurotransmitter with a comparable time scale. To test this, we have changed the decay time constant of INP release in a wide range (from 10⁻⁵ to 10⁻³ms⁻¹) and checked that this population can still regulate LC activity.

6. Simplified network model

We now propose a mean-field like model, based on simplified Wilson-Cowan’s equations, that can reproduce the firing rates of all three populations with their corresponding mechanisms of information transmission. We intend to provide a simplified tool that can be related to the realistic simulations and be used for further analysis.

First, we propose the following system of coupled non-linear differential equations for all populations:

$$\frac{{dF}_j}{dt}(t)=A_j{\theta }_j\left[I(t)b_j+\sum _{k=0}^3{a}_{jk}{F}_k(t)\right]-{\gamma }_0[F_j(t)-F_{j0}],$$ (5)

where F_j(t) is the firing rate of population j (with 0 →HCRT, 1 →GABA, 2 →LC and 3 →INP), θ_j being a population-specific gain function, A_j a rate of excitation, γ_j a damping term and F_j0 the residual activity when there is no excitation. I(t) is the external current only applied to the HCRT group (i.e., b_j = δ_j,0). We have used as gain function θ_j(x) = 1/2{tanh[(x − β_j)/σ_j] + 1}.

To find the appropriate coefficients for each population separately, we use a least-square minimization and neglect some small coefficients (see Appendix B), ending up with the following equations

$$\frac{{dF}_0}{dt}(t)=A_0{\theta }_0[I(t)b_0]-{\gamma }_0[F_0(t)-F_{00}],$$ (6)

$$\frac{{dF}_1}{dt}(t)=A_1{\theta }_1[a_{10}{F}_0(t)]-{\gamma }_1[F_1(t)-F_{10}],$$ (7)

$$\frac{{dF}_2}{dt}(t)=A_2{\theta }_2[a_{10}{F}_0(t)+a_{21}{F}_1(t)+a_{22}{F}_2(t)+a_{23}{F}_3(t)]-{\gamma }_1{F}_2(t),$$ (8)

$$\frac{{dF}_3}{dt}(t)=A_3{\theta }_3[a_{30}{F}_0(t)+a_{32}{F}_2(t)]-{\gamma }_3[F_3(t)-F_{30}].$$ (9)

In Fig. 6, the LC parameters were fitted resulting in A₂ = 20.1, γ₂ = 0.8, a₂₀ = 4.98, a₂₁ = 4.16 and a₂₂ = −0.36. Additional values of the fitted coefficients are described in the Appendix B.

Figure 6.

Mean-field Equations (6–9) after parameters were fitted using a least-squares minimization (see Appendix B for details). Bright lines are simulation results with same parameters for comparison. In this particular case, we have fitted the mean-field model using a steady current applied from t = 60s on (similar to Fig. 3[c]). This is shown in the left panel. On the right panel, we have used the parameters from this fitting and used the model with a square-pulse current (same as used in Fig. 3[a]). This shows that the mean-field equations captures the trends of all populations and can be used as a simplified view of this hypothalamic network.

To further testing our model, we compare the solutions of these mean field equations with simulations when the parameters are different than what have been used to fit them. For instance, changing the amplitude of the current used in the protocol described by Fig. 1, our simplified model can reproduce simulations within an error of 20%. We show in Fig. 6 a case in which we have fitted the model with a protocol close to Fig. 3[c] and then used it to predict the behavior of the system with a current square pulse as in Fig. 3[a]. The errors may vary in the range 5–30%.

These simplified equations capture the trends of all populations within reasonable agreement, giving a more intuitive insight on how these population interact with each other.

7. Discussions

In this paper we have modeled a neural system composed of three populations of two-compartment conductance-based model neurons: the HCRT cells, composed of neurons expressing the HCRT neuropeptides, the LC cells, responsible for triggering the sleep-wake transition, and the GABA interneurons, which inhibit both HCRT and LC neurons through GABA_A receptors. Although direct optogenetic stimulation of LC cells leads to wakefulness, vigilance state transitions are not exclusively managed by this neural group. LC neurons integrate two completely different time scales: one of the order of milliseconds, used by AMPA and GABA neurotransmitters, and another of the order of seconds, used by the HCRT neuropeptides. We have thoroughly analyzed the mechanism of LC activity regulation, which should be capable of suppressing overloads of activity that lead to behavioral states similar to episodes of neuropsychiatric disorder [21].

We have shown that due to GABA_A neuroreceptors the activity of LC cells actually increases. This unexpected effect is a result solely of how these cells respond to inhibitory pulses. An analysis of their Phase-Resetting curves (PRCs) shows that 80% of the phase range leads to an advance in the upcoming spike times, which thus leads to an increase in firing rate. This is due to the fact that the LC cell model integrates actively its inputs, in contrast to passive models as the leaky integrate-and-fire model. Thus a fast hyperpolarization pulse consequently provokes a fast rising in the membrane potential and an advance in spike timing.

This effect of increased activity induced by GABAergic neurons have been previously observed, although through different mechanisms [49]. Also an inhibitory synapse that causes spike timing advances is not completely rare, and we can cite for instance bursting neurons in central pattern generators [50].

Since a fast inhibition would not consistently regulate LC activity, we propose the existence of a population that releases a neuropeptide similar to the hypocretin/orexin but with an inhibitory effect – i.e., working in a slower time scale. A simplified view of the new network configuration is depicted in Fig. 7. This population, which we refer to as INP, could use for instance any known inhibitory neuropeptide with a long enough time scale. We have shown that with this new model the LC activity can be regulated, especially if there is a feedback connectivity: not only INP inhibits the LC cells, but also the LC excites back the INP. We have proposed this connectivity since a large activity coming from the LC population would immediately evoke a larger inhibition by INP neurons. By carefully choosing the parameters, we have shown that this can actually lead to a nimble rise up of LC activity and, when the stimulation of HCRTs is over, a similarly swift end of activity. This is probably the most compelling argument that the existence of a neuropeptide with such characteristics could be of great importance to this hypothalamic circuit. Among the possible INPs acting on LC neurons are MCH [51–53] and opioid [54, 55] peptides. Finally we proposed a mean-field model that captures the main element in the interaction among those groups and can be used as a simplified view of this hypothalamic network for further analysis.

Figure 7.

New configuration proposed for an active control of the LC activity. We have introduced a population which release a slow neuropeptide inhibiting LC group. We propose a feedback connection from LC to INP for a more robust control mechanism. Among possible INPs acting on LC neurons we can cite MCH and opioid peptides.

We have also reported that a slower GABAergic inhibition, similar to what is usually called GABA_A,slow currents, may also decrease the LC activity. However, this kind of current is unlikely to happen in LC cells and the regulation is less stable than the results achieved by this hypothetical INP population.

The interplay of two neuropeptides balancing each other have been studied before as well, though from different perspectives [47, 48, 56]. Experimental tests can be performed in this hypothalamic circuit to find out if our predictions are true, especially starting by active stimulation (by current clamp, for instance) of these GABAergic cells and studying the effect on the LC population – this should induce an increase of firing rate. This empirical observation, regardless of the ending result, may bring new ingredients for understanding the role of each of these three populations and how LC activity is regulated.

Appendix A. GABAergic interneuron model details

We provide here additional details to section 2 of the currents used in Equations 1 and 2. Time dependence and units are intentionally dropped to simplify the notation whenever they are clear in the context.

We start with the currents involved in the axon compartment. First the sodium current, modeled as I_Na = g_Nam²n(V_A − 50mV) [57], with g_Na = 312nS and

$$v_1=18\text{mV}+V_t-V_A,v_2=-40\text{mV}-V_t+V_A,V_t=-55\text{mV},$$ (A.1)

$$\stackrel{.}{m}=\frac{0.32v_1}{e^{v_1/4}}(1-m)-\frac{0.28v_2}{e^{v_2/4}}m,$$ (A.2)

$$v_3=17\text{mV}+V_t-V_A,v_4=40\text{mV}+V_t-V_A,$$ (A.3)

$$\stackrel{.}{n}=0.128e^{v_3/18}(1-n)-\frac{4}{1+e^{v_4/5}}n.$$ (A.4)

Next, the delayed rectifier potassium current was implemented as I_Kd = g_Kdh(V_A + 60mV), with g_Kd = 96nS

$$v_1=35\text{mV}+V_t-V_A,v_2=20\text{mV}+V_t-V_A,$$ (A.5)

$$\stackrel{.}{h}=\frac{0.016v_1}{e^{v_1}}(1-h)-\frac{0.25v_2}{e^{v_2/40}}h.$$ (A.6)

The potassium dependent on calcium is I_K[Ca] = g_K[Ca]h(V_A+60mV), with g_K[Ca] = 20nS and

$$\mathrm{\Gamma }(x,y,z)=\frac{1}{1+exp\left(\frac{x-y}{z}\right)},$$ (A.7)

$$[\stackrel{.}{C}a]={10}^{-3}\left(-0.35I_{Ca}-{\mu }^2\{[Ca]+0.04\}\right),$$ (A.8)

$$\stackrel{.}{h}={10}^{-3}\mathrm{\Gamma }(0.08,[Ca],0.011)(1-h)-0.002h,$$ (A.9)

where we have set the calcium dynamics dissipation μ = 1.5 and I_Ca is the calcium current, which takes effect into the soma compartment, described next.

The calcium current was modeled using the Goldman-Hodgkin-Katz formalism [58] as I_Ca = g_Cal³/(1 − e^2V_s/24.42), with g_Ca = 1.76nS and

$$\stackrel{.}{l}=0.1[\mathrm{\Gamma }(-V_s,43,0.5)-l].$$ (A.10)

Finally, the transient potassium current is modeled as I_A = g_Ah(V_S + 60mV), with g_A = 500nS and

$$\stackrel{.}{h}=\frac{\mathrm{\Gamma }(-V_S,41,1)-h}{350-349\mathrm{\Gamma }(-V_S,-48,4)}.$$ (A.11)

For simulating these populations, one can access an implementation published in Model DB (accession number 145162) of the network with HCRT and LC cells [16, 59].

Appendix B. Fitting the mean-field model

Equation 5 defines a class of models from which we want to find one that reproduces our numerical results. The idea then is to fit from data A_j, a_jk, β_j, σ_j and γ_j for all j, k ∈ {0, 1, 2, 3}. Since F_j0 are easily accessible from simulations and in order to decrease the complexity of the fitting process, we have neglected them and assumed the values given below. We use a simple procedure of minimization of an least-squares objective function. First, for clarity we shall use the following notation F_j (t; A_j, a_j, β_j, σ_j, γ_j), with a_j = (a_j0, a_j1, a_j2, a_j3), to explicitly express the parameters dependence by the mean field Equation 5. Suppose is the firing rate time series of population j, for instance as shown in Fig. 2. In order to make the convergence of the parameters during the minimization easier, we have at this point used a Gaussian Smoothing algorithm in these time series. Then we can estimate the derivative by calculating

$${\mathcal{F}}_j^{\prime }(t)=\frac{{\mathcal{F}}_j(t+\mathrm{\Delta }t)-{\mathcal{F}}_j(t+\mathrm{\Delta }t)}{\mathrm{\Delta }t}.$$ (B.1)

In most of examples, Δt = 0.05ms. We then construct the following objective function,

$$\mathcal{O}(A_j,a_j,{\beta }_j,{\sigma }_j,{\gamma }_j=\sum _t{\left[{\stackrel{.}{F}}_j(t;A_j,a_j,{\beta }_j,{\sigma }_j,{\gamma }_j)-{\mathcal{F}}_j^{\prime }(t)\right]}^2.$$ (B.2)

Then we substitute Equations (6–9) into Equation (B.2). Finally, the solution is given by arg min_{Aj,aj,βj,σj,γj} { (A_j, a_j, β_j, σ_j, γ_j)}, which then is found by a Broyden-Fletcher-Goldfarb-Shanno algorithm.

However, to actually use F_j (t; A_j, a_j, β_j, σ_j, γ_j) in Equation (B.2) we need the firing rate of other populations. This is a omitted dependence and we have used, only during the minimization process, the time series to supplement this need.

Given the number of parameters, we have removed some of them in order to make the analogy to Fig. 1 as clear as possible. For instance, a₀₂ was set to zero because there is no direct contribution from the LC group to HCRT cells. Also other parameters were removed from the final mean field equations (6–9) because they were found to be too small, thus only adding more noisy and iterations to the algorithm. This is the case of a₀₁, representing the feedback from GABA to HCRT.

Finally, we provide the parameters fitted for Fig. 6. For HCRTs, A₀ = 20.0, γ₀ = 3.56, β₀ = 1.4, σ₀ = 0.8, F₀₀ = 2.4 and b₀ = 1.0. For GABAs, A₁ = 6.93, γ₁ = 0.5, β₁ = 2.16, σ₁ = 1.0, F₂₀ = 2.7 and a₁₀ = 0.84. Notice that F_j0 agrees with the residual firing rate of each population.