Excitation of an agonist muscle and simultaneous inhibition of its antagonist is necessary for efficient force generation during movement. This is Sherrington's principle for reciprocal innervation. For example, when we flex an elbow, the flexor group of muscles receives excitatory impulses from the corresponding neurons. The same neurons also inhibit neurons innervating the antagonist muscle group, the extensors. Figure 1A schematically illustrates this phenomenon. The green lines are excitatory neural projections and red lines are inhibitory. Reciprocally inhibitory neural circuits are present in many central areas responsible for limb movement 9. Furthermore, some limb movement related neurons also exhibit a rebound increase in their firing rate when inhibition is removed – post-inhibitory rebound (PIR). For example, PIR is observed in neurons of premotor areas for limb movements, such as the thalamus and inferior olive 10.

Under physiological circumstances, with normal levels of external inhibition, PIR provides sufficient force to generate prompt, high-speed ballistic movements. Reciprocal innervation and PIR also make neural circuits inherently unstable and prone to oscillations 1112. Figure 1, panels B and C, illustrate why a circuit of reciprocally innervated neurons with PIR is prone to oscillations. A neural circuit is formed by two mutually inhibitory neurons (neuron-A and -B). A small pulse of neural activity, either from spontaneous fluctuations in neural activity or a small spontaneous movement, activates neuron-A. The discharge from neuron-A inhibits neuron-B. Following the termination of the input to neuron-A, its discharge drops and inhibition upon neuron-B is removed. Neuron-B in turn shows PIR and starts to fire. Since neuron-B also inhibits neuron-A, when firing in neuron-B stops, PIR occurs in neuron-A. Thus the reciprocally-innervated neural circuit begins to oscillate (Figure 1C).

Nature's solution to this inherent instability in a reciprocally innervated neural circuit is to add enough external inhibition to keep the excitability of the constituent neurons under control. For example, we have proposed that oscillations in an analogous circuit controlling ballistic eye movements (saccades) are normally prevented by tonic external glycinergic inhibition 12. Similarly, abolishing GABAergic inhibition in GABA mutant mice is known to cause tremor 13. However, oscillations in reciprocally innervating circuits can be seen in the presence of normal external inhibition. For example, in patients with ET GABA mutation was not found 14. In the following section we introduce a concept explaining the basis of oscillations in reciprocally innervated circuits when external inhibition is intact.

Oscillations in reciprocally innervated circuits could emerge if the relative effect of intact external inhibition is reduced by an increased excitability within the reciprocally-innervated neurons themselves. Therefore it is possible that increased neural excitability can overcome the effects of normal external inhibition. There could be a number of causes of increased excitability including an increase in either the hyperpolarization activated cation current (I_h) or the low threshold calcium current (I_T) 1516 or alterations in the intracellular levels of second messengers and the regulators that influence the activation kinetics of these ion channels 161718.

We recorded postural limb tremor in 22 patients with ET, and used a neuromimetic model to simulate their tremor. We tested our hypothesis by simulating a Hodgkin-Huxley type, conductance-based, neuromimetic model of pre-motor burst neurons responsible for ballistic limb movements. This model has the following features: (1) A circuit consisting of reciprocally innervating excitatory and inhibitory neurons. (2) Physiologically-realistic membrane kinetics of the premotor neurons determined by specific and physiologically plausible subsets of membrane ion channels. The latter also determines the excitability of the membrane. (3) These model neurons had a property of rebound firing after sustained inhibition - post-inhibitory rebound (PIR). By increasing specific membrane conductances that are known to increase PIR and neural excitability, such as I_h and I_T, we could simulate the range of frequencies of tremor recorded from ET patients.

We studied 22 ET patients, who gave written, informed consent before enrolling in the study.

Here we summarize the key features of our computational model. Readers are referred to additional files 1, 2 and 3 for more details on methodology of computational simulations. The Hodgkin-Huxley equations were implemented to simulate action potentials. GABA and glycine mediated inhibitory chloride conductance was implemented for inhibition, and non-NMDA and NMDA glutamatergic channels for excitation. Kinematics of CaV3 channels (carrying I_T) and four subtypes of HCN channel (HCN1-HCN4; carrying I_h) were included to simulate PIR and to modulate neuronal excitability. The activation kinetics of each of the subtypes of ion channels carrying I_h are different, HCN-1 being the fastest and HCN-4 the slowest, with the others having intermediate activation time constants 20.

The following equation describes the time evolution of the membrane potential of the brain stem neurons:

C*dV/dt = -I_L - I_T - n1I_h1- n2I_h2- n3I_h3- n4I_h4- I_Na - I_K - I_Cl - I_NMDA - I_nonNMDA

where V is the membrane potential of the neuron, C is the membrane capacitance (1 μF/cm2) and n1-n4 is a rate scaling factor determining the ion channel expression profile in the neuron. I_L, I_T, I_h1-4, I_Na, and I_K, denote the leak current, low-threshold calcium current, hyperpolarization activated current (carried by HCN1-4), fast sodium current and delayed rectifier potassium current, respectively. I_Cl is the synaptic current mediated by glycinergic and GABAergic neurotransmitters. I_NMDA and I_nonNMDA are synaptic currents mediated by NMDA and non-NMDA sensitive glutamate receptors. The parameters n1-n4 represent the relative expression strength of I_h channels. The model simulated a set of four burst neurons – inhibitory agonist, inhibitory antagonist, excitatory agonist, excitatory antagonist. We assumed a small, yet physiologically plausible, variability in the ion channel expression profiles of the simulated burst neurons, which in turn, accounted for minor (physiological) variability in their membrane properties. Supplementary figure 1 (additional file 2 ) schematizes the model organization, synaptic weights, and mathematical functions implemented in the model. Additional details of the model are in additional file 1 as well as in Miura and Optican (2006) and Shaikh, et al., (2007).

We simulated membrane properties of reciprocally-innervated burst neurons within a local feedback loop model for ballistic movements. Each neuron was a conductance-based single-compartment model. The major conductances simulating the membrane properties of the model neurons are schematized in Figure 2A. The details of the model organization and the mathematical equations driving these ion currents are explained in methods and additional file 1, 2 and 3. The model is compatible with the known anatomical organization of neural circuits for limb movements. The membrane properties of the model burst neurons are also consistent with the known profiles of limb-movement sensitive pre-motor neurons. The model simulated normal ballistic limb movements when the ion currents and excitability of the simulated burst neuron membranes were within physiological limits (I_h = 0.1 mSeimens; I_T = 2.0 mSeimens; and resting membrane potential = -68 mV). Increases in I_h and I_T further depolarized the resting membrane potential producing increased neural excitability (Figure 2B). Oscillations resembling ET were simulated when I_h and I_T in the model neurons were increased. The increase in these currents resulted in alternating bursts of action potentials in the neurons innervating the sets of agonist and antagonist muscles (Figure 2C). In Figure 2C the membrane potential is plotted along the y-axis and time along the x-axis. The alternating bursts of discharge reflect the circuit oscillations and produce the limb tremor. The simulated oscillations (the model output) are illustrated as the grey trace in Figure 2D. The common time scale in Figure 2C and 2D facilitates the comparison of the simulated oscillations (model output) with the alternating bursts of discharge in the pairs of neurons innervating agonist and antagonist muscles. The black trace in Figure 2D illustrates a representative postural limb tremor recorded from one ET patient. The simulated tremor superimposes fairly well on the postural limb tremor from the ET patient. The frequency of the simulated tremor (5.9 Hz) is close to the actual tremor frequency (5.7 Hz). The correlation coefficient between the two waveforms was 0.9.

Postural tremor in ET has a relatively wide range of frequencies 21. Therefore, we asked if the conductance-based membrane model could simulate the inter-subject variability in the tremor frequency. The range of frequencies and corresponding amplitude of the postural tremor from 22 ET patients are illustrated in Figure 3A. The tremor frequency in these ET patients ranged from 3–11 Hz. We simulated this variability in frequency by changing the value of I_h and I_T. How the amplitude and frequency of tremor depend on the value of I_h and I_T in our model is shown in Figure 3B and 3C. In these figures the x- and y-axes represent I_T and I_h, respectively. The oscillation frequency (Hz) and amplitude (degree) are color coded and plotted along the z-axes. Tremor occurred when conductance through I_h was larger than 1 mS/cm² (approximately 10 times larger than its physiological value, 0.1 mS/cm²; S: seimens) 2223. The effects of increase in I_h on changes in frequency and amplitude, for a given constant value of I_T, were investigated. Increasing I_h, while keeping I_T at a constant value, increased the frequency of tremor (slope ± 95% confidence interval: 0.5 ± 0.05). However, when I_T was kept constant, there was only a slight decrease in the tremor amplitude when I_h was increased (slope ± 95% confidence interval: -0.02 ± 0.10). Then we investigated the changes in the frequency and amplitude of tremor when I_T was increased but I_h was kept at a constant value. With a constant I_h, the tremor frequency decreased when I_T was increased (slope ± 95% confidence interval: -3.9 ± 0.1). In a similar analysis, the amplitude of tremor, however, increased with increasing I_T (slope ± 95% confidence interval: 1.1 ± 0.01).

If I_h was increased above the physiological values, ET was simulated for any I_T that was 2.75 mS/cm² or larger (normal I_T: 2.0 mS/cm², reference 26). In contrast, if I_h was near its physiological range, increasing I_T alone did not simulate oscillations.

We tuned the model to match the tremor frequency measured in each of the 22 ET patients. Then we plotted the observed versus simulated frequency for each patient as colored diamonds, indicating the model values of I_h and I_T required for that patient (Figure 4). There was a nearly perfect correlation between the frequency of simulated tremor and the corresponding postural tremor in the patients (all the data points fall along the black dashed equality line). The data points in Figure 4 are color-coded according to the value I_T or I_h. The higher values of I_h correspond to ET patients with higher tremor frequencies (Figure 4A). Consistent with the results in Figure 3, the range of I_h to simulate ET was 10 – 80 times larger than its normal physiological value. The corresponding range of I_T to simulate the characteristics of ET was only 1.3 – 3 times higher than its physiological value. Our results suggest that the variability of tremor frequency among ET patients is determined by both I_h and I_T. Tremor frequency increases with increasing I_h and decreases with increasing I_T.

In summary, simulations support our hypothesis that an increase in premotor neural excitability, caused by increasing I_h and I_T, results in oscillations of a reciprocally innervated neural circuit. These oscillations resemble ET. The frequency of tremor is determined by both I_T and I_h.

Increases in membrane excitability could affect motor control in many ways. One mechanism is by destabilizing the circuits comprised of reciprocally innervated neurons. Such circuits exist at many levels in the central nervous system. This pattern of reciprocal innervation between pre-motor neurons projecting to a pair of agonist-antagonist muscles is fundamental for efficient force generation during ballistic movements 9. The stability in these circuits requires adequate external inhibition. Either removal of external inhibition 12 or increasing the excitability of the constituent neurons (as proposed here) could lead to oscillations that produce tremor.

We used a conductance-based model of burst neurons with physiologically realistic membrane properties and anatomically realistic neural connections to test this hypothesis. The cardinal features of this model were threefold: 1) increased neural excitability secondary to increase in I_h and/or I_T, 2) post-inhibitory rebound (PIR), and 3) inherent circuit instability resulting from reciprocal innervation between the neurons projecting to agonist and antagonist muscle pairs. The model simulated oscillations resembling ET when I_h was increased (with or without an increase in I_T). While suggesting an overall conceptual framework underlying oscillatory behavior, our model can not pinpoint the specific anatomical neural networks that produce ET. Nevertheless, we can reasonably suggest that the following anatomical regions might be involved.

Circuits in the thalamus, inferior olive, cerebrum and cerebellum are involved in the generation of limb movements. One mechanism for tremor is that a group of neurons within a single nucleus develops an abnormal oscillatory mode. In this mode, a neural discharge is followed by a prolonged hyperpolarization that terminates in rebound spikes. Thus, each neuron oscillates independently. Synchronization of such independently oscillating neurons could result in rhythmic activity that becomes strong enough to cause gross motor oscillations. Electrotonic coupling through connexin gap junctions can facilitate synchronization in premotor nuclei such as the inferior olive 27. Cells of the inferior olive also express ion channels that carry I_h and I_T. Moreover, I_h is thought to influence the synchronization of oscillations in the inferior olive 28. It is possible that increased intracellular levels of cAMP in the inferior olive increases I_h conductance to facilitate these oscillations. Octanol, which reduces synchronized oscillations in the inferior olive, also reduces ET 2930.

Reciprocal innervation occurs in many neural circuits controlling limb movements 912. For example, thalamo-cortical relay (TC) neurons send glutamatergic excitatory projections to thalamic reticular (TR) neurons. TR neurons send GABA mediated inhibitory projections to TC neurons 313233. TR neurons mutually inhibit each other via inhibitory collaterals 313233. Thus interactions between TC and TR neurons make reciprocal loops with positive feedback from the TC to TR neurons, negative feedback from TR to TC neurons, and mutually inhibitory TR neurons 34. The globus pallidus internus (GPi) sends inhibitory projections to TC and TR neurons in the motor thalamus 3536. Furthermore, thalamic and GPi neurons carry I_h and I_T, and exhibit post-inhibitory rebound 1153738. Hence, the reciprocally innervated neural network formed by the TC and TR neurons might be inherently unstable 12. Physiologically, this network is under control by external inhibition from the GPi.

Bursting behavior in neurons, which is fundamental for making a circuit prone to oscillations, is also seen in the subthalamic nucleus 39. Reciprocal innervation – a key feature for oscillating neural circuits – is also apparent in the spinal cord 9. Thus there are a number of areas related to motor control within the central nervous system that are comprised of neurons and circuits that are prone to oscillations.

In our model we increased the I_h to increase membrane excitability and produce oscillations. As this is a lumped model, we can not differentiate among increasing the maximal conductance of an individual channel, increasing the number of channels, or increasing the probability that a channel is open. Any combination of these changes would increase I_h. Likewise, there are a number of intracellular modulators regulating I_h including intracellular levels of calcium, cAMP, and pH 1516 that could affect membrane excitability. Again, our simulated lumped model cannot tell us which of these factors might be the cause. We had to increase I_h by ten-fold to simulate tremor. Although the range of pathological changes in channel currents is not known, a ten-fold increase does not seem unreasonable, as experimental studies have shown that conductances can be changed several fold 17. Thus, the conceptual underpinning of our lumped model seem plausible.

Naturally, we do not expect that all patients with ET have the same cause for their increased excitability and consequent tremor. But, with the wide range of ion channels that participate in the molecular cascade that accounts for PIR, many types of alterations could change membrane excitability and in turn lead to the phenotype of ET. Similarly, the effects of loss of inhibition from the cerebellar Purkinje neurons (which may be abnormal in some patients with ET, see below) might also change the membrane threshold and increase the excitability of the premotor neurons, and lead to the appearance of tremor in some ET patients 2526. Indeed, our model underscores the importance of external inhibition in preventing oscillations in a reciprocally innervated circuit. For example, removal of external inhibition in an analogous model accounted for limb tremor in a syndrome called micro-saccadic oscillation and limb tremor – microSOLT 12.

Some ET patients have the gly9 susceptibility variant of the DRD3 gene 4041. DRD3 receptor is expressed in the thalamus and substantia nigra where limb-movement related neurons are present 42. Two of the possible effects of this mutation could be directly related to the regulation of membrane excitability. There is a prolonged intracellular action of mitogen-activated protein kinase (MAPK) in the gly9 variant of DRD3 40. The latter can cause increased intracellular levels of cAMP via excessive inhibition of phosphodiesterase E4 434445. Conversely, it was also shown that gly9 variant in DRD3 gene is associated with reduced forskolin induced formation of cAMP 40. Although it is not known what the intracellular levels of cAMP in ET are, it is possible that they are altered. If they are higher, I_h could be increased and thus lead to an increase in membrane excitability.

Our simulations suggested that increasing I_h, and thus increasing excitability, increases the tremor frequency. It was also reported that the frequency of ET decreases and the amplitude increases with age 46. Aging might alter the profile of membrane ion channels; our model explains the effects of maximal ion conductance on the tremor frequency. Furthermore, our simulations suggest that a parallel increase in I_T, when I_h is already increased, reduces the frequency of the tremor (Figure 3B) and increases its amplitude (Figure 3C). This explanation does not exclude the possibility that aging could also change other membrane and circuit properties including the latency of the long feedback loop around the burst neurons.

Although our hypotheses remain to be proven experimentally, they suggest new approaches to understanding common tremor disorders. The conceptual scheme that we present here could also be used to analyze tremor disorders other than ET. However, the unique aspect of this model is that it can simulate oscillations by changing intrinsic membrane properties of the burst neurons and does not require any 'structural' changes in the anatomical organization or connectivity of the constituent neurons. The latter is relevant to ET, since there are no gross structural changes except in some cases in which there is a decrease in cerebellar Purkinje neurons 2526. Loss of external inhibition to reciprocally innervating circuits could be an important underlying pathophysiological mechanism component in other tremor disorders including Parkinson's disease, cerebellar tremor, and micro-saccadic oscillations and limb tremor 1234.

Finally, treatment with pharmacological blockers targeted towards ion channels may offer therapeutic benefits. For example, although counterintuitive, interfering with the function of a normal ion channel to decrease membrane excitability in the face of impaired external inhibition might reduce oscillatory behavior. Such an approach is similar to that for some forms of inherited epilepsy in which seizures are presumably caused by an abnormal ion channel. Treatment then can target another, presumably intact channel 47.