Abstract

We simulate the inhibition of Ia-glutamatergic excitatory postsynaptic potential (EPSP) by preceding it with glycinergic recurrent (REN) and reciprocal (REC) inhibitory postsynaptic potentials (IPSPs). The inhibition is evaluated in the presence of voltage-dependent conductances of sodium, delayed rectifier potassium, and slow potassium in five -motoneurons (MNs). We distribute the channels along the neuronal dendrites using, alternatively, a density function of exponential rise (ER), exponential decay (ED), or a step function (ST). We examine the change in EPSP amplitude, the rate of rise (RR), and the time integral (TI) due to inhibition. The results yield six major conclusions. First, the EPSP peak and the kinetics depending on the time interval are either amplified or depressed by the REC and REN shunting inhibitions. Second, the mean EPSP peak, its TI, and RR inhibition of ST, ER, and ED distributions turn out to be similar for analogous ranges of G. Third, for identical G, the large variations in the parameters’ values can be attributed to the sodium conductance step () and the active dendritic area. We find that small on a few dendrites maintains the EPSP peak, its TI, and RR inhibition similar to the passive state, but high on many dendrites decrease the inhibition and sometimes generates even an excitatory effect. Fourth, the MN's input resistance does not alter the efficacy of EPSP inhibition. Fifth, the REC and REN inhibitions slightly change the EPSP peak and its RR. However, EPSP TI is depressed by the REN inhibition more than the REC inhibition. Finally, only an inhibitory effect shows up during the EPSP TI inhibition, while there are both inhibitory and excitatory impacts on the EPSP peak and its RR.

1.  Introduction

In the past few years, the physiological and morphological properties of the cat and rat -motoneurons have been intensively studied (Bui, Grande, & Rose, 2008a, 2008b; Heckman, Hyngstrom, & Johnson, 2008; Hyngstrom, Johnson, & Heckman, 2008; Jean-Xavier, Pflieger, Liabeuf, & Vinay, 2006; Larkum, Rioult, & Luscher, 1996). This revealed a large number of voltage-dependent conductances on motoneuron (MN) dendrites, uniquely located relative to other types of neurons (Heckmann, Gorassini, & Bennett, 2005; Kim, Major, & Jones, 2009; Kuo, Lee, Johnson, Heckman, & Heckman, 2003; Saint Mleux & Moore, 2000). In contrast to hippocampal and neocortical pyramidal cells, which show a variety of different channel distributions depending on the neuron and channel type (see Magee, 1998) in MNs some dendrites are active, while others remain passive (Luscher & Larkum, 1998). The voltage-dependent channels and the depolarizing shunting inhibition provide the excitatory potential not only for input amplification or attenuation but also for nonlinear interactions.

In our previous research, we analyzed the influence of active MN's dendritic properties on the EPSP (Gradwohl & Grossman, 2008) and IPSP (Gradwohl & Grossman, 2010). We concluded that voltage-dependent channels on the dendrites boost EPSP peak and diminish IPSP peak. However the mean EPSP and IPSP peaks of step function (ST), exponential rise (ER), and exponential decay (ED) distributions are similar when the range of G is equal. Our model is a continuation of our previous simulations. We analyze the effective modulation of the EPSP by the IPSPs in the presence of the same basic voltage-dependent channels and explore any relation between the type of voltage-dependent channel distribution in the dendrites and the EPSP inhibition.

Gain modulation is used for integrating data from different sensory, motor, and cognitive sources. At the neuronal level, gain modulation can be seen as a nonlinear summation of the synaptic response of excitatory synapses containing a voltage-dependent parameter (Mel, 1993; Volman, Levine, & Sejnowski, 2010).

There are two types of motor neuron inhibition: the orthodromic reciprocal inhibition (REC) and the antidromic Renshaw inhibition (REN). The latter is part of a feedback system of the -MN and operates within a polysynaptic interneuronal pathway (Schneider & Fyffe, 1992). The REC inhibitory inputs are distributed on the soma and proximal dendrites (Burke, Fedina, & Lundberg, 1971; Curtis & Eccles, 1959; Stuart & Redman, 1990) and the REN inhibitory inputs on the dendrites distally to the soma (Fyffe, 1991; Maltenfort, McCurdy, Phillips, Turkin, & Hamm, 2004), similar to the excitatory AMPA and NMDA inputs. Additionally, the time course of the REC IPSP compared to the REN IPSP is shorter. Thus, the presence of the REN and REC inhibitory inputs in the simulation serves as a basis for analyzing the effect of voltage-dependent channels on differently located synapses with distinct conductance time course. Glycine (not GABA) is the major neurotransmitter involved in the inhibitory pathway of the spinal cord (Capaday & van Vreeswijk, 2006) in REC inhibition. Although REN inhibition employs glycine and GABA neurotransmitters (Cullheim & Kellerth, 1981; Jonas, Bischofberger, & Sankuhler, 1998) we neglected the latter in order to compare REC and REN inhibition with the same glycinergic neurotransmition. The ionic current of the inhibitory synapses is carried by chloride, and, as a result, the reversal potential is close to the resting potential (Gao & Ziskind-Conhaim, 1995; Jean-Xavier, Mentis, O'Donovan, Cattaert, & Vinay, 2007; Jean-Xavier et al., 2006; Stuart & Redman, 1990; Yoshimura & Nishi, 1995). It is known that in mature motoneurons after spinal cord injury (Boulenguez et al., 2010; Edgerton & Roy, 2010) and in a newborn rat (Marchetti, Pagnotta, Donato, & Nistri, 2002), the inhibitions have depolarizing characteristics. Inhibition is mainly attained by a shunting effect. It is regularly mentioned as a mechanism by which gain modulation is controlled through changes in the input resistance, without affecting the membrane potential (Prescott & De Koninck, 2003). Shunting inhibition modulates synaptic excitatory depolarization, whereas hyperpolarizing inhibition has a subtractive effect on the depolarization. The inhibition in our model has both depolarizing and shunting properties that are similar to our previously published model (Gradwohl & Grossman, 2010). Consequently, it is of special interest to examine its interaction with the depolarizing excitatory synaptic potentials since the depolarizing inhibition occurs in mature motoneurons after spinal cord injury (Boulenguez et al., 2010; Edgerton & Roy, 2010) and also in newborn rats (Marchetti et al., 2002).

In this work, the inhibition of Ia-EPSP synaptic potentials is evaluated in the presence of the basic voltage-dependent components: sodium, delayed rectifier potassium, and slow potassium currents distributed at different locations on the dendrites. Although persistent inward currents (PIC) are a dominant conductance on the MN's dendrites (Heckman et al., 2008), we do not include them in our investigation, as we are interested in the basic active components in the absence of neuron firing (see Luscher & Larkum, 1998). REN and REC inhibitions were inserted into the model and mediated by the glycinergic neurotransmission. We then analyzed the interaction of excitatory and inhibitory inputs between themselves and the voltage-dependent channels. In addition, we incorporated the dendritic branching (Cullheim, Fleshman, Glenn, & Burke, 1987a, 1987b; Segev, Fleshman, & Burke, 1990) and the MN's size. As a result, we were able to simulate realistic morphological MNs and compare their distinct physiological properties.

The gain modulation in our research is not evaluated by the firing rate of the neurons (Mitchell & Silver, 2003; Prescott & De Koninck, 2003) or by the altering of the frequency-input current slope (Higgs, Slee, & Spain, 2006), but rather by the tuning of the subthreshold potentials. The inhibitory conductance affects subthreshold potentials and may be influenced by various parameters, such as the distribution functions of its synaptic inputs and the voltage-dependent conductances across the dendritic tree. Moreover, the strength of the voltage-dependent conductance and the inhibitory synaptic conductance may determine the resultant response of the Ia-EPSP. We simulated through detailed compartmental models five morphologically (Cullheim et al., 1987b) and physiologically (Fleshman, Segev, & Burke, 1988) characterized triceps surea MNs of different types and addressed the following questions:

  1. Which kind of inhibition depresses the Ia-EPSP components of peak, time integral, and rate of rise most efficiently?

  2. How does an active dendrite contribute to the efficacy of inhibition? Alternatively, which kind of dendritic voltage-dependent channels distribution depresses Ia EPSP most effectively?

  3. Is Ia-EPSP inhibition monotonically dependent to the input resistance of the MNs?

2.  Methods

2.1.  Computer Model.

2.1.1.  Morphological and Physiological Parameters.

The morphological and physiological data for the analysis of five - motoneurons (MNs) of the muscle triceps surea were taken from adult anesthetized cat MNs (Segev et al., 1990; see www.neuromorpho.org). We also derive the input resistance (RN), the membrane time constant (, and the surface area (Am) from the intracellular recordings by Cullheim et al. (1987b). In their work, they reconstructed the three-dimensional morphological structure of the -MNs with the aid of a light microscope after a color reaction with injected horseradish peroxidase (HRP) (Cullheim et al., 1987a, 1987b). The model simulated one slow (S), two fast fatigable (FF), and two fast fatigable resistance (FR) types of - MN. Finally, we refer to earlier studies by Burke and Glenn (1996) for the numbers of the synapses and their spatial distribution on the dendrites.

2.1.2.  Simulation and Passive Membrane Properties.

We define the MNs according to a compartmental model (Jones & Bawa, 1997; Segev et al., 1990; Vieira & Kohn, 2007) and run the simulations on an IBM- Lenovo PC using the software NEURON (Hines, 1989). Each segment is assumed to be isopotential with an electrical length smaller than 0.2 . We set the number of compartments of MN 36, MN 38, MN 41, MN 42, and MN 43 as 559, 739, 777, 854, and 731, respectively. We rely on the model by Traub (1977) for the morphological data for the initial segment, the myelinated axon, and the node of Ranvier. The modeled and values are then matched to the experimental values by selecting appropriate specific membrane resistance (Rm), specific cytoplasm resistance (Ri), and specific membrane capacity (Cm) (Fleshman et al., 1988). We set Rm to be low at the soma and high at the dendrites using a step model and vary the ratio of somatic Rm and dendritic Rm for each cell according to Fleshman et al. (1988). Specifically, Ri was set at 70 cm, Cm at 1 F/cm2, and Vrest at −70 mV. Each -MN contains an axon with a particular characteristic: axon hillock: L = 100 m, d = 10 m, Cm= 1 F/cm2; axon with myelin: L = 400 m, d = 16 m, 0.05 F/cm2; node of Ranvier: L = 75 m, d = 20 m, C 1 F/cm2. Finally, we apply a time step of 10 s for the simulations.

2.1.3.  Synaptic Input.

We include in the model 300 excitatory inputs of AMPA (Segev et al., 1990) and NMDA receptors, located at the same segments (Moore, Buchanan, & Murphey, 1995). In accordance with Redman and Walmsley (1983), the conductance of the AMPA receptor is simulated by an - function with a maximal conductance (gmax) of 5 nS and a time to maximal conductance ( of 0.22 ms. For the conductance of NMDA receptors, we follow the convention of our previous study (Gradwohl & Grossman, 2008, 2010).

The reversal potential (Vrev) of the synaptic potentials, activated by glutamate at the AMPA receptors, is 0 mV (Engberg & Marshall, 1979; Segev et al., 1990) similar to that at the NMDA receptors (Brodin et al., 1991). The physiological delay of the synaptic activation time is between 0 and 1.4 ms (Segev et al., 1990). The synaptic response, generated by the model through the activation of the two types of receptors, is compared to the experimental results of Jahr and Yoshioka (1986), who demonstrated a significant contribution of the NMDA components in slowing the EPSP decay time. The density distribution function of the excitatory synapses at the dendritic tree is assumed to follow a gaussian function with maximal values at a distance of 200 to 400 m relative to the soma (Burke & Glenn, 1996; Cullheim et al., 1987a, 1987b; Ornung, Ottersen, Cullheim, & Ulfhake, 1998).

In addition, we incorporate 70 postsynaptic glycinergic reciprocal (REC) and Renshaw (REN) inhibitory synapses into the model. The number of the synapses involved in the REC inhibition is based on the observation that an average of 69 Ia-interneurons is in contact with a single MN (Jankowska & Roberts, 1972). As has been shown in earlier work (Brown & Fyffe, 1981; Fyffe, 1991), every contact usually yields two or three synaptic buttons. However, if the probability of the glycinergic neurotransmitter release is smaller than one assuming 0.5 (Jack, Redman, & Wong, 1981), it will yield approximately 70 active synapses. We determine a similar number for the synapses of the REN inhibition. We set the REC inhibitory synapses on the proximal dendrite near the MN soma (Curtis & Eccles, 1959) and assume exponential behavior of the density distribution of the REC inhibition as a function of the distance from the soma (see Figure A1 in Gradwohl & Grossman, 2010). A gaussian function is assumed to mimic both the REN inhibition (see Figure A1 in Gradwohl & Grossman, 2010) and the EPSP density distribution (Gradwohl & Grossman, 2008; Segev et al., 1990). We distribute the activation time of the inhibitory synaptic activation within a time window of 0 to 1.4 ms (Luscher, Ruenzel, & Henneman, 1979) and set the maximum conductance associated with the inhibitory postsynaptic potentials (IPSP) at 9.1 nS (Stuart & Redman, 1990). In contrast to Segev et al. (1990), we do not compute the REN and REC inhibitions as -functions but as a composition of two exponential functions, analogous to previous work by Schleifstein-Attias, Tarasiuk, and Grossman (1994) and Gradwohl and Grossman (2010). The reversal potential (Vrev) is defined for the mature MNs following injury and in “healthy” immature MNs at 5 mV above the resting potential, indicating depolarizing chloride channels (Gao & Ziskind-Conhaim, 1995; Jean-Xavier et al., 2006, 2007; Stuart & Redman 1990; Yoshimura & Nishi, 1995). Finally, we set the of the REC and REN conductances at 7 ms and 18 ms, respectively.

2.2.  The Numerical Model.

We simulate the voltage-dependent channels through the H&H equations (Hodgkin & Huxley, 1952):
formula
2.1
where is the synaptic current, a the cable's radius, and Ri the specific intracellular resistance. Iion is the voltage-dependent current defined by
formula
2.2
where gNa is sodium conductance, gKf is fast potassium conductance, gKs is slow potassium conductance, and gleak is leak conductance while:
formula
2.3
and
formula
2.4

The constants in equation 2.4 are taken from adult cat -motoneurons (Traub, 1977). This is a common practice; Destexhe, Neubig, Ulrich, and Huguenard (1998) determined the dynamics of thalamic conductances using hippocampal neurons data, and Luscher and Larkum (1998) adjusted the parameters of the organotypic cultured motoneurons’ conductances using information from neocortical pyramidal cells.

2.3.  Types of Simulation Models.

In our simulations, we assume that the voltage-dependent channels are located only on the dendrites and are distributed according to one of three specific functions. The simulations deal with subthreshold potentials based on the premise that even subthreshold depolarizing IPSPs may activate multiple voltage-dependent conductances (Andreasen & Lambert, 1999; Gonzalez-Burgos & Barrionuevo, 2001; Stuart & Sakmann, 1995), similar to what is observed in EPSP (Gradwohl & Grossman, 2008; Magee, 1998). The three voltage-dependent channel functions are analogous to a fast inactivating Na+ conductance (gNa) (Barrett, Barrett, & Crill, 1980) and both a fast (gKf) and a slow (gKs) potassium conductance (Barrett et al., 1980) that seems to be gKCa (Viana, Bayliss, & Berger, 1993). Spike repolarization and AHP are mainly influenced by gKf and gKs, respectively (Schwindt & Crill, 1981). There are most likely other currents on the MN's initial segment, the soma and the dendrites, which undoubtedly shape the subthreshold potentials and the firing properties (Viana et al., 1993). These are, for example, persistent sodium (INap), calcium current producing a plateau potentials (Bui, Cushing, Dewey, Fyffe, & Rose, 2003), or multiple K+ channels, like IKCa. However, our goal now is to analyze the effective modulation of the EPSP by the IPSPs in the presence of the most basic voltage-dependent channels, such as gNa, gKf, and gKs, that affect both the EPSP (Gradwohl & Grossman, 2008) and IPSPs (Gradwohl & Grossman, 2010). In order to simplify the model, we neglect Ca2+ channels and simulate the IKs only by voltage-dependent conductance with no Ca2+ dependency (see Luscher & Larkum, 1998). Locating voltage-dependent conductances on the dendrites in the presence of a passive soma and axon increases the EPSP peak (Gradwohl & Grossman, 2008) while the IPSP peak decreases (Gradwohl & Grossman, 2010) relative to a full passive neuron. In contrast to the channels in the cortical pyramidal neurons (Stuart & Sakmann, 1995), MN's voltage-dependent channels are sometimes absent in certain dendrites while appearing in neighboring ones (Larkum et al., 1996). We located the dendritic voltage-dependent channels according to Larkum et al. (1996), who determined that motoneurons’ active channels are not distributed equally along dendritic tree branches. Furthermore, the density distributions of voltage-dependent conductances in the active dendrites are actually unknown. This led us to select three types of channel distributions (Gradwohl & Grossman, 2008), defined at a distance of 0 to 400 m from the soma (see Larkum et al., 1996, Figure 5) and overlapped only partly the REC and REN synapses: (1) a step function (ST) with uniform active conductance density over the dendritic tree; (2) an exponential decay (ED), where high-conductance density, located proximal to the soma, decays exponentially away from the soma; and (3) an exponential rise (ER), where proximal low-conductance density increases exponentially with distance. Densities of the sodium conductance (gNa) are varied relative to the type of conductance distribution between a minimum and a maximum value in order to attain equal total conductance, S, Siemens) for each MN. The densities of the delayed rectifier (gK) and slow (gKs) potassium conductances were one-third relative to gNa (Luscher & Larkum, 1998). According to Jones and Bawa (1997), the gKf and gKs at some MN types are similar.

For each density model, we simulate with two, four, six, and eight active dendrites. The total number of the dendrites of MN 36, MN 38, MN 41, MN 42, and MN43 is 10, 13, 12, 8, and 11, respectively. In each case, we execute 10 runs of randomly selected dendrites.

3.  Results

3.1.  Determination of gNa_ step and G.

In the first stage of our research, we focus on determining the and G for comparing the distinct models. We accomplish this by varying the numbers of active dendrites and the along the three selected models ST, ED, and ER (see section 2). However, none of this information is experimentally available for MNs. In previous simulation work, we had assumed that all MNs contain the same voltage-dependent conductance density (gActive) on the dendrites (0–400 m) near the soma (Gradwohl & Grossman, 2004). As a result, the affected membrane area was shown to differ between the MNs and yield varying total active G for each MN and each model. This made any comparison between the simulations problematic. In Gradwohl and Grossman (2008, 2010) and in this investigation, we adopt a different approach and keep the range of G equal for all MNs. We accomplish this by dividing the precalculated minimal and maximal for each MN into five steps and then simulating the resultant EPSPs (Gradwohl & Grossman, 2008) and IPSPs (Gradwohl & Grossman, 2010).

In order to perform simulations on MNs at the same G range of the dendrites at all types of voltage-dependent channels distribution, the in the ED model has to be set higher than in the ER model due to the higher surface area in the branching distal dendrites than proximal dendrites. In the ST model, has to be the smallest one. The uncertainty about the population of active dendrites in each MN also forces us to simulate 10 random dendrite combinations for 2, 4, 6, and 8 dendrites at the five (see section 2). This statistical approach yields 200 values of PSP (EPSP + IPSP) components as a function of G for each model (e.g., Figure 3) for every MN. This enables us to compare the results of the ST, ER, and ED models at the same MN and between different MNs. In Figure 1B, for example, the distinct somatic and dendritic PSPs and EPSPs as a function of are shown. Furthermore, as represented in Figure 1C, the voltage-dependent currents decrease the PSP peak and its decay time compared to the passive PSP.

Figure 1:

ED, MN42. Somatic and dendritic EPSPs (only excitatory synapses are activated) and PSPs (activation of inhibitory and excitatory synapses) with eight active dendritic trees. (A) Recurrent PSPs at a voltage-dependent conductance of gmin + 2 . versus time with their corresponding EPSP. Superimposed simulations; the delay of the PSP indicates the time interval between the IPSP and the initiation of the EPSP. (B) Superimposed and arbitrarily shifted recurrent PSPs and EPSPs versus at a time interval of 8 ms. (C) Dendritic passive and active recurrent PSPs versus time at a time interval of 8 ms and gmin. IK and INa slightly decrease the active PSP peak and its decay time in comparison to the passive PSP.

Figure 1:

ED, MN42. Somatic and dendritic EPSPs (only excitatory synapses are activated) and PSPs (activation of inhibitory and excitatory synapses) with eight active dendritic trees. (A) Recurrent PSPs at a voltage-dependent conductance of gmin + 2 . versus time with their corresponding EPSP. Superimposed simulations; the delay of the PSP indicates the time interval between the IPSP and the initiation of the EPSP. (B) Superimposed and arbitrarily shifted recurrent PSPs and EPSPs versus at a time interval of 8 ms. (C) Dendritic passive and active recurrent PSPs versus time at a time interval of 8 ms and gmin. IK and INa slightly decrease the active PSP peak and its decay time in comparison to the passive PSP.

3.2.  Time Course of Inhibition of EPSP's Peak, Time Integral, and Rate of Rise.

In order to reveal the time course of each inhibition on the excitatory input, the REN and REC IPSPs (Gradwohl & Grossman, 2010) are initiated at various time intervals (TIV) prior to the EPSP (see Figure 1A) and recorded at the soma and a dendritic location. However, in the following figures, we record the responses only at the soma since all the synaptic currents coalesce there. As we described in section 2, we simulate all types of MNs and voltage-dependent channel distributions and assess the efficacy of the inhibitory process as a percentage change in the PSP peak, the time integral (TI), and the rate of rise (RR) relative to the EPSP alone according to the fraction (EPSP − PSP)/EPSP as shown for MN 42 in Figure 2. Recall that the inhibitory chloride conductances inserted in the model produce mainly shunt inhibition with a depolarizing IPSP. Therefore, the behavior of the PSP peak, TI, and RR as a function of TIV is biphasic: at short TIV, we observed a prominent inhibitory effect, while at longer TIV, the depolarizing inhibitory potential augments the EPSP components at both REN and REC inhibitions. At short TIV, the EPSP peak is depressed by REC and REN inhibitions on the order of 15% and 9%, respectively (see the upper graph of Figure 2). In contrast, at long TIV, the EPSP peak is enhanced by both depolarizing inhibitions. As expected from their conductance duration, the EPSP TI is depressed for longer periods by the REN inhibition than by the REC inhibition (see the left lower graph of Figure 2). However, the maximal EPSP TI inhibition is similar (about 30%) for both inhibitions. Interestingly, the EPSP peak and RR inhibitions (see the right lower graph of Figure 2) during all tested TIV are similar for REC (max 15%) and REN (max 9%) inhibitions. As before, the duration of the effective enhancing of the EPSP RR was longer for the REN inhibition due to the difference in decay time of the inhibition (Gradwohl & Grossman, 2010)

Figure 2:

REC and REN inhibitions of EPSP's peak, time integral (TI), and rate of rise (RR) of MN 42. The MN contains voltage-dependent channels ED model distribution at all eight dendrites. Positive values of inhibition denote the actual decrease of the parameter and negative values denote excitation.

Figure 2:

REC and REN inhibitions of EPSP's peak, time integral (TI), and rate of rise (RR) of MN 42. The MN contains voltage-dependent channels ED model distribution at all eight dendrites. Positive values of inhibition denote the actual decrease of the parameter and negative values denote excitation.

3.3.  Effects of gNa_ step and G on the Time Course of REN and REC EPSP Inhibition.

The inhibition of EPSP parameters depends on the TIV between the activation of the inhibitory pathway and the initiation of the EPSP. The results for EPSP peak, TI, and RR modifications by the REN and REC inhibitions, as a function of G at three TIVs—0 ms, the shortest; 8 ms, the most effective; and 32 ms, the longest for MN42—are shown in Figure 3.

Figure 3:

The REC and REN inhibitions of the EPSP peak, time integral (TI), and rate of rise (RR) versus G are shown at three significant time intervals (TIV: 0, 8, and 32 ms) at the MN 42 ED model. At 8 ms, TIV EPSP was depressed most effectively. Enhancing the diminishes the inhibitory effect and increases the excitatory one. Positive values of inhibition denote the actual decrease of the parameter, and negative values denote excitation.

Figure 3:

The REC and REN inhibitions of the EPSP peak, time integral (TI), and rate of rise (RR) versus G are shown at three significant time intervals (TIV: 0, 8, and 32 ms) at the MN 42 ED model. At 8 ms, TIV EPSP was depressed most effectively. Enhancing the diminishes the inhibitory effect and increases the excitatory one. Positive values of inhibition denote the actual decrease of the parameter, and negative values denote excitation.

3.3.1.  EPSP Peak Inhibition.

The upper graph in Figure 3 exhibits examples of the relationship between EPSP peak inhibition and the total G activated for each run of the 200 simulations. Considering the scatter of data points, there does not seem to be any statistically significant difference between the REN and REC inhibitions with respect to the ability to depress the EPSP peak. Yet a closer examination of each cluster reveals some interesting features. Increasing , as can be seen in the gray coded forms for various , decreases EPSP peak inhibition. The minimal data points are found at the top, and the maximal g points are at the bottom of the cluster. A clear example of such a distribution can be seen in the REC EPSP peak inhibition of the ED model in Figure 3. Thus the vertical scatter (bottom-up) of the data is due in part to the decrease in . The horizontal scatter (left to right) of the data points is caused by the increased number of active dendrites (larger G) in the simulations. It is important to note that even identical G activation levels may yield variable inhibition of EPSP components, an effect that is dependent on both the size of and the area of active dendrites.

This means that choosing a large number of active dendrites with small improves the inhibition of EPSP compared to fewer active dendrites with high For example, at G of 100 S in the ED model with TIV of 8 ms (see Figure 3), the EPSP peak REC inhibition with the minimal and eight active dendrites is about 7.5%, while the range of this inhibition at the same G with the maximal and two active dendrites falls between −4% and −1% (excitatory effect). Figure 4 shows the EPSPs and PSPs for the same G attained by two different combinations of s and number of active dendrites. The highest G is obtained only with combinations of maximal and the highest number of dendrites (8 out of a total of approximately 12), which decreases the chance for repetitions in the lottery.

Figure 4:

Two reciprocal MN42 ED PSPs at a time interval of 8 ms at two different s with their control EPSPs at the same G of about 100 S. Inhibitions at the maximal and minimal are −3.19 % (excitation) and 7.14 %, respectively. Therefore uniformly distributed voltage-dependent channels increase inhibitions, while densely located ones decrease inhibition and sometimes generate even an excitatory effect.

Figure 4:

Two reciprocal MN42 ED PSPs at a time interval of 8 ms at two different s with their control EPSPs at the same G of about 100 S. Inhibitions at the maximal and minimal are −3.19 % (excitation) and 7.14 %, respectively. Therefore uniformly distributed voltage-dependent channels increase inhibitions, while densely located ones decrease inhibition and sometimes generate even an excitatory effect.

REN and REC inhibitory synapses generate similar EPSP peak inhibitions at the three chosen time intervals. The proximally located REC synapses, relative to the soma, yielded similar EPSP peak inhibition at all TIVs in comparison to the distally distributed REN. Since the inhibitions in our model have a characterization of depolarizing inhibition, we observe an inhibitory effect during the shunt, or short TIVs. In contrast, after the termination of the inhibitory conductance and at longer TIVs, an excitatory effect becomes prominent, as can be seen by the negative values on the y-axis curve. At both types of inhibition, on 8 ms time interval yields the most effective inhibition, whereas at a time interval of 32 ms, the excitatory effect becomes dominant. Despite the apparent variations, the mean EPSP peak REC and REN inhibition are similar for each MN (see Figure 6). Surprisingly, the EPSP peak inhibition is saturated very quickly as the number of active dendrites (G) is increased for each g. This means that only a relatively small part of the dendritic tree (estimated 4% to 10%) is active when effective inhibition of the EPSP peak occurs.

3.3.2.  EPSP TI Inhibition.

EPSP TI inhibition due to the REC synapses at the 0 and 32 ms TIVs reveals an excitatory effect. Especially at 32 ms TIV, the maximal excitation is approximately 30% (indicated as a −30% inhibition). However, at the 8 ms TIV, the mean EPSP TI inhibition is only approximately 15%. The REN inhibition is shifted upward compared to the REC inhibition on the y-axis. As a consequence, the maximal and minimal inhibitions of EPSP TI at 8 ms TIV are about 30%. We again see that the relationship of the EPSP TI inhibition to the and G is similar in comparison to the EPSP peak and RR inhibitions (see below).

3.3.3.  EPSP RR Inhibition.

The EPSP RR is again affected by the TIV and not by the type of the inhibitory synaptic input. For example, EPSP RR depression by the REN synapses at 0, 8, and 32 ms TIVs ranges between −5% and −20%, 10% and 5%, and −5% and −23%, respectively. We again see that the cluster of the EPSP RR inhibition is dependent on the and G in similar fashion to that of the EPSP peak and TI inhibition.

3.4.  Efficacy of EPSP Inhibition at Various Active Dendritic Models.

The relationship between the EPSP REN and REC inhibitions as a function of the total activated G and for each run of the 200 simulations for MN42 at 8 ms TIV is shown in Figure 5. General examination of the whole cluster of data points shows no significant difference between the ST, ED, and ER active dendrite models with respect to the ability to inhibit the EPSP peak and its kinetics parameters TI and RR.

Figure 5:

The REC and REN inhibitions of the EPSP peak, time integral (TI), and rate of rise (RR) versus G were demonstrated at the most significant 8 ms time intervals (TIV) at the MN 42 ST, ED, and ER distribution models. For all models, an inhibitory effect was attained. At this MN, REN inhibition was more effective in inhibiting EPSP peak and TI relative to REC inhibition. However, statistically (see Figure 6) the efficacy of REN and REC inhibitions at the five MNs is similar. Positive values of inhibition denote actual decrease of the parameter, and negative values denote excitation.

Figure 5:

The REC and REN inhibitions of the EPSP peak, time integral (TI), and rate of rise (RR) versus G were demonstrated at the most significant 8 ms time intervals (TIV) at the MN 42 ST, ED, and ER distribution models. For all models, an inhibitory effect was attained. At this MN, REN inhibition was more effective in inhibiting EPSP peak and TI relative to REC inhibition. However, statistically (see Figure 6) the efficacy of REN and REC inhibitions at the five MNs is similar. Positive values of inhibition denote actual decrease of the parameter, and negative values denote excitation.

3.4.1.  EPSP Peak Inhibition.

The maximal inhibitions are similar in ST, ED, and ER models for analogous combinations of small with small G (“min” in figure legends). However, in the REN inhibition, the maximal excitation (“max” in figure legends) is smaller relative to the REC inhibition. For example, the REC inhibition in the ST model ranges between 12.5% and −7%, while the REN inhibition in the same active model is between 15% and 1%. In all active models, the combination of the small with small G inhibits the EPSP peak most effectively. However, the mean EPSP peak inhibition is similar for REN and REC inhibitions at all active models (see Figure 6).

3.4.2.  EPSP TI Inhibition.

EPSP TI is depressed by REC and REN inhibitions at all , resulting in the lack of any excitatory effect. REC and REN inhibitions decrease the EPSP TI from 11% to 17% and 26% to 32%, respectively. Note that the range of EPSP TI inhibition at both inhibitions is 6%. The different efficacy of inhibition is present at all MNs, not only for MN 42 (see Figure 6).

3.4.3.  EPSP RR Inhibition.

Similar to the EPSP peak inhibition, the EPSP RR is depressed by REC and REN inhibitions approximately 10% at the minimal . At maximal , the REN inhibition depresses EPSP RR by only 5%, while REC leads to an even less effect and becomes excitatory in the ER model. Furthermore, we observe that the REN inhibition leads to a smaller range (smaller standard deviation) of the EPSP RR depression at all MNs relative to the REC inhibition (see Figure 6). The clusters of the EPSP RR inhibition are similar to those at the EPSP peak inhibition.

Figure 6:

The mean and the SD of EPSP peak, time integral (TI), and rate of rise (RR) inhibitions obtained for the five MNs for the two inhibitory pathways using the three simulating models: ER, ED, and ST distributions. The inhibition efficacy is shown for the maximal and minimal at a time interval (TIV) of 8 ms. Note that the type of voltage-dependent channels dispersion does not influence the inhibition of the EPSP.

Figure 6:

The mean and the SD of EPSP peak, time integral (TI), and rate of rise (RR) inhibitions obtained for the five MNs for the two inhibitory pathways using the three simulating models: ER, ED, and ST distributions. The inhibition efficacy is shown for the maximal and minimal at a time interval (TIV) of 8 ms. Note that the type of voltage-dependent channels dispersion does not influence the inhibition of the EPSP.

3.5.  Is EPSP Inhibition Efficacy Related to the MN's Input Resistance?.

All simulations are performed at the five MNs that differ in their input resistance (RN). In an attempt to explore the relationship between the inhibition efficacy and the MN's RN, we plot the EPSP peak, the TI, and the RR inhibitions from each cell for the two inhibitory pathways using the three simulating models ER, ED, and ST of the active channels distributions on the dendrites (see Figure 6). The inhibition efficacy is calculated for the maximal and minimal at a time interval of 8 ms. It is important to emphasize that the mean effect is statistically significant since its standard deviation is generally small. It seems that MNs’ RN does not affect the inhibitory processes at any simulation model. However, for each active model, the minimal improves the EPSP peak, TI, and RR inhibitions in comparison to the maximal . Interestingly, the REC and REN shunting inhibitions yield an inhibitory or an excitatory effect on the EPSP peak as a function of the , whereas EPSP TI is inhibited by both inhibitions. REC inhibition affects the EPSP RR as a function of : at small ones, the effect is inhibitory, while at larger ones, it is excitatory. However, REN inhibition produces an inhibitory effect on the EPSP RR. We show statistically that REN and REC similarly alter the EPSP peak and RR in the range of 10% to −10%. Yet EPSP TI is differentially decreased: the REN inhibition falls into the range of 25% and 45%, whereas the REC inhibition lies between 10% and 35%.

4.  Discussion

In this study we insert two types of depolarizing inhibitions into the MN model: the orthodromic reciprocal inhibition (REC), located proximal to the soma, and the distally distributed Renshaw inhibition (REN), which depresses Ia EPSP via a feedback pathway. We also introduce three types of voltage-dependent channels: density distributions of transient sodium, delayed rectifier potassium, and slow potassium dispersed on the dendrites near the soma (0–400 m) of five reconstructed - MNs. This allows us to evaluate their effects on the inhibition of the EPSP amplitude, the TI, and the RR.

Our major conclusions are:

  1. EPSP peak and its kinetics are either amplified or depressed by the REC and REN shunting inhibitions depending on the TIV between the inhibitory and the excitatory inputs.

  2. The mean EPSP peak, TI, and RR inhibitions of the three distribution models are similar for equal ranges of G.

  3. The variations in the parameters values for identical G depend on the and active dendritic area. Small on a few dendrites generates an EPSP peak, its TI, and RR inhibition like the passive state, but high on many dendrites decreases the inhibition and generates at some s even an excitatory effect.

  4. A small area of active dendrite, or 4% to 10% of the total dendritic area, does not change the efficacy of inhibition compared to the passive state.

  5. The MN's input resistance does not alter the efficacy of EPSP inhibition.

  6. While the REC and REN inhibitions lead to similar modifications of the EPSP peak and RR, the REN inhibitions affect the EPSP TI significantly more strongly than the REC inhibitions.

  7. At the time interval of 8 ms, the EPSP TI is always inhibited by REC and REN pathways, whereas EPSP peak and RR are bimodally modified in an inhibitory and an excitatory manner.

  8. The presence of sodium and delayed rectifier potassium conductances is enough to balance the efficacy of REC and REN inhibitions of EPSP parameters, despite the different location of the inhibitory synapses. However, in a passive model (Rall 1967), the inhibition due to proximally located inhibitory conductance was the most effective one relative to distal ones.

Our simulation tries to mimic the inhibition of mature cat motoneurons following injury. We adopt the morphological data from adult MNs (Cullheim et al., 1987a, 1987b; Fleshman et al., 1988) and the dispersion of the dendritic voltage-dependent channels from MNs in organotipic culture (Luscher & Larkum, 1998). Obviously this strategy yields some inaccuracies. Neurons in organotipic culture may acquire some morphological and physiological modifications due to the artificial environment relative to their in vivo counterparts.

4.1.  The Function of Depolarizing Shunting Inhibition.

We analyze the Ia-EPSP peak as a function of the TIV between the activation of the inhibitory depolarizing IPSP and the excitatory input. As a result, we are able to establish two distinct phases. During the short TIV, inhibition suppressed PSP peak (and consequently action potentials generation), whereas during the longer TIV, when the inhibitory conductance was terminated, the IPSP caused an excitation due to the summation of the IPSP depolarizing tail and the EPSP depolarization. Similarly, Jean-Xavier et al. (2007) showed that depolarizing IPSPs could either inhibit or facilitate EPSPs depending on the relative timing and location of the PSPs. Schleifstein-Attias et al. (1994) have also tested the glycinergic inhibition of spinal monosynaptic reflex (MSR) in newborn rat while GABAergic inhibition was eliminated by the use of biccuculine. When they applied a conditioning stimulus to the adjacent dorsal root at different TIVs prior to the activation of the MSR, they observed a similar two-phase behavior of the inhibition. In their case, however, they could not exclude the involvement of some facilitation that could lead to the action potential firing of the MNs. The inhibitory effect in their experiments could therefore even be underestimated. While the relative contribution of each of the MSR modulation components is unclear, our results indicate that the second phase of IPSP synaptic excitation may result purely from summation of IPSP and EPSP. Another possible source of IPSP excitation in neonatal rat spinal cord could be the Ih current (Kjaerulff & Kiehn, 2001). This current is located in dendrites, slowly activated by hyperpolarization, noninactivating, and is carried by mixed cations whose reversal potential is positive to the resting potential. Thus, this current may contribute to the formation of a long-lasting excitation. In this model, we neglect these types of currents and insert only the most basic ones. Thus, we limit the number of unknown variables like the size of the maximal active conductance and the accurate distribution function on the dendrite.

In this study, we analyzed the subthreshold potentials in order to understand the basis of synaptic potential interactions. We did not simulate spiking at this stage. However, several previous studies have shown that shunting inhibition might modify spiking activity. The firing pattern is commonly described by either the firing rate of the neurons or its firing rate gain, defined as the slope of the relation between input current to a neuron and its firing rate (F-I curve). Holt and Koch (1997) claimed that the shunting inhibition does not modify the firing rate gain but shifts the F-I curve on the F-axis. Similarly, Brizzi et al. (2004) reported that shunting inhibition reduced the firing frequency without changing the firing rate gain. Capaday and van Vreeswijk (2006) revealed that the current from the dendrite to the soma was attenuated by the shunting inhibition and thus reduced the firing rate gain.

The shunting inhibition in mature MNs following injury and in newborn -motoneurons is mediated by chloride conductance with a reversal potential above the resting potential. This excitatory behavior has an interesting effect on spike initiation and plateau potential, as reported at neostriatal neurons (Flores-Barrera et al., 2009). The plateau potential can affect the firing pattern. However, the recurrent network at the neostriatum consists of inhibitory neurons. The authors showed that in addition to glutamatergic excitatory synaptic currents, as well as intrinsic L-type Ca2+ currents, inhibitory synapses help to maintain slow depolarization while depressing firing at the beginning of the response. According to our study, depolarizing inhibition in the MN could amplify the excitatory potentials and thereby initiate action potentials during the late phases when conduction increase is being terminated.

4.2.  Similar Efficacy of REN and REC Inhibitions.

In his classic work, Rall (1967) simulated the relationship between inhibitory and excitatory inputs on a single dendrite, based on a passive 10-compartment model. The inhibition was located at three different sites, whereas the excitation remained at the same compartment. He demonstrated that the inhibition due to the proximally located conductance was the most effective one, with its magnitude inversely related to its distance from the soma. However, in our MN simulations, the presence of sodium and potassium conductances and the absence of Ca2+ channels (Carlin, Bui, Dai, & Brownstone, 2009) were enough to balance the efficacy of REC and REN inhibitions of EPSP parameters, despite the different location of the inhibitory synapses. The same phenomena can be attained by the presence of Ca currents on the dendrites. Solinas, Maex, and De Schutter (2006) demonstrated through a passive model simulation that the amplitude and time course of IPSPs recorded at the soma were dependent on the synapse distance from the soma. However, after the addition of active Ca2+ channels, the effect of distance on the amplitude and width of the IPSP was significantly reduced, with no effect on its rise time. Similarly, through a computer model of a cerebellar Purkinje cell, which included dendritic voltage-dependent Ca2+ channels, De Schutter and Bower (1994) were able to show that small distally located excitatory inputs evoked a response at the soma of similar size to those generated by more proximal synapses. In contrast, the same model containing only passive parameters exhibited a significantly smaller current of the distal inputs than in the proximal case. We can therefore conclude based on the above models that the firing pattern is insensitive to the exact location of the synaptic input.

4.3.  Efficacy of Low Conductance Density of Voltage-Dependent Channels.

We found that in order to preserve EPSP inhibition similar to the passive case, the conductance density of the voltage-dependent channels, which is located on the dendrites, should be low. However, conductance that is too high results in less effective inhibition. This is due to the fact that a high density of voltage-dependent channels generates a spike-like response, which is more difficult to inhibit. However, the total densities of the voltage-dependent channels on the dendrite and axon are similar since the surface area of the dendrite is about 100 times larger than the surface of the axon (Curtis & Appenteng, 1993), while the active conductance densities are inversely related (Luscher & Larkum, 1998). Thus, if the MN is normally utilizing a large nonlinear contribution of dendritic voltage-dependent current to its EPSP (Poznanski & Bell 2000), it can, at the same time, lead to a powerful postsynaptic inhibition. Based on our previous work, the REC PSP peak inhibition in a passive model seems to be about 14% in MN 42 (Gradwohl, Nitzan, & Grossman, 1999) like the inhibition in a model with low dendritic active conductance density, as attained from this letter. This small level of voltage-dependent channel density is important not only for maintaining effective inhibition like the passive state, but also for having proper action potential backpropagation, as has already been described for neocortical neurons (Stuart & Sakmann 1994). The low dendritic sodium conductance density allows only antidromic spikes, arriving from the soma, to initiate dendritic spikes. Computer simulations in these cortical neurons (Koch, Rapp, & Segev, 1996) suggested dendritic gNa of 4 mS/cm2 in comparison to a “hot” axon having times 40 greater density. Similar behavior was found in cultured -motoneurons (Larkum et al., 1996). Somatic and dendritic patch electrodes separated by a distance of 30 to 423 m recorded spontaneous synaptic activity and evoked action potentials at the soma. At all times, the action potential was initiated first at the soma (or proximal to the soma) and only later antidromically activated the dendritic active response. In order to explain these experimental results, Luscher and Larkum (1998) reconstructed a realistic computer model. They concluded that the dendrites and the axon of these cultured - MNs had to contain gNa at 3 mS/cm2 and 700 to 1200 mS/cm2, respectively. In fact, increasing the dendritic active conductances of the neocortical neuron and the cultured -MN might delay the spike from backpropagating, while the action potential continues to spread from the soma to the axon (Mackenzie & Murphy, 1998). Our results from the realistic MN model therefore corroborate these experimental and modeling results. They confirm the importance of the low density of voltage-dependent conductance in the dendritic tree.

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