## Abstract

Neural information is characterized by sets of spiking events that travel within the brain through neuron junctions that receive, transmit, and process streams of spikes. Coincidence detection is one of the ways to describe the functionality of a single neural cell. This letter presents an analytical derivation of the output stochastic behavior of a coincidence detector (CD) cell whose stochastic inputs behave as a nonhomogeneous Poisson process (NHPP) with both excitatory and inhibitory inputs. The derivation, which is based on an efficient breakdown of the cell into basic functional elements, results in an output process whose behavior can be approximated as an NHPP as long as the coincidence interval is much smaller than the refractory period of the cell's inputs. Intuitively, the approximation is valid as long as the processing rate is much faster than the incoming information rate. This type of modeling is a simplified but very useful description of neurons since it enables analytical derivations. The statistical properties of single CD cell's output make it possible to integrate and analyze complex neural cells in a feedforward network using the methodology presented here. Accordingly, basic biological characteristics of neural activity are demonstrated, such as a decrease in the spontaneous rate at higher brain levels and improved signal-to-noise ratio for harmonic input signals.

## 1. Introduction

Neural activity is characterized by sets of spiking events that travel within the brain through neuron junctions that receive, transmit and process stream of spikes (Rieke, Warland, van Steveninck, & Bialek, 1997; Kandel, Schwartz, & Jessell, 2000). These spiking events are stochastic in nature, starting at the peripheral end (Kiang, 1965; Teich & Khanna, 1985; Teich, Heneghan, Lowen, Ozaki, & Kaplan, 1997), and reach higher processing levels (e.g., Abeles, 1982).

The stochastic properties of the auditory neural system have been studied since the 1950s. Spontaneous neural activity was best described as a homogeneous point Poisson process provided that refractory time was ignored (Rodieck, Kiang, & Gerstein, 1962). Further investigations of the auditory system revealed a description of the neural response as a nonhomogeneous Poisson point process (NHPP) whose instantaneous rate depends on the input stimuli (Rieke et al., 1997; Gray, 1967). These studies ignored the discharge history including the refractory period. Other studies tested the effect of discharge history on depletion in response rate (Gaumond, Kima, & Molnar, 1983) while others modeled the neural response as a doubly stochastic fractal point process (Teich, 1989) or as modified NHPP (Johnson & Swami, 1983; Carney, 1993).

Analysis of the stochastic properties of each level in the brain and of each type of neural cell is essential for characterizing the neural system behavior as a function of input stimuli. In this letter, we concentrate on fundamental coincidence detectors (CD) that receive independent excitatory and inhibitory inputs and generate a spike if the number of excitatory inputs exceeds the number of inhibitory inputs by a known number during a short interval.

In the auditory system, coincidence detectors cells are found in several brainstem nuclei. Such cells are mainly involved in estimating interaural differences, which are the primary acoustical cues for sound localization. They were found in both the superior olivary complex (SOC) and the inferior colliculus (IC) (Rose, Brugge, Anderson, & Hind, 1967; Goldberg & Brown, 1969; Yin & Chan, 1990; McAlpine, Jiang, Shackleton, & Palmer, 1998; Park, 1998; Joris, Smith, & Yin, 1998; Agmon-Snir, Carr, & Rinzel, 1998; Smith, Owens, & Forsythe, 2000; Palmer, Shackleton, & McAlpine, 2002).

The initial processing of interaural timing cues occurs in the medial superior olive (MSO) in which neurons receive excitatory input from the large, spherical, bushy cells of both (right and left) antro ventral cochlear nucleus (AVCN), which preserve, and even enhance, the timing accuracy seen in the auditory nerve, providing exquisitely timed inputs (Warr, 1966, 1969; Palmer et al., 2002; Joris, Carney, Smith, & Yin, 1994). These types of cells are frequently called excitatory-excitatory (EE) cells (e.g., Guinan, Guinan, & Norris, 1972; Joris et al., 1994).

Another type of CD cell is found in the lateral superior olive (LSO), where neurons receive inhibitory inputs from neurons in the ipsilateral medial nucleus of the trapezoid body (MNTB), which in turn receive excitatory input from the globular bushy cells of the contralateral cochlear nucleus (Glendenning, Hutson, Nudo, & Masterton, 1985; Cant, 1991; Warr, 1972; Boudreau & Tsuchitani, 1968). The pathway from AVCN to MNTB is characterized by synapses producing secure, short-latency responses and therefore near-coincident arrival at the LSO of the ipsilateral excitation and the contralateral inhibition. Those neurons are sensitive to the balance of intensity at the ears because the excitation due to ipsilateral sounds is reduced by increasing levels of contralateral sounds (Boudreau & Tsuchitani, 1970; Guinan et al., 1972; Caird & Klinke, 1983; Caspary & Finlayson, 1991). Those cells are frequently referred as excitatory-inhibitory (EI) cells (e.g., Guinan et al., 1972; Tollin & Yin, 2002).

Existing computational methods for neural cells analysis can explain the functionalities of a general CD cell. Several computational methods are biological constraints that can be used to investigate the type of information emerging from within the brain's neural activity in neural cells (Herz, Gollisch, Machens, & Jaeger, 2006). For example, in integrate-and-fire models (reviewed by Burkitt, 2006a, 2006b), the cell neural membrane potential is obtained by applying a decision rule. Whenever the membrane potential crosses a given threshold, the cell generates spikes. A complete neural net might include a thousand or more cells, and Monte Carlo simulation can yield a description of the neural system (e.g., Amemori & Ishii, 2001).

Another way to characterize the neural system behavior as a function of input stimuli is by analyzing the stochastic properties of each level in the brain and of each type of neural cell. The output of such a system can be used to predict known behavioral properties. Such a computational method was used for the auditory nerve where just noticeable differences (JND) of a tone's frequency, level, and interaural cues were predicted (Siebert, 1968, 1970; Colburn, 1973; Stern & Colburn, 1978; Heinz, 2000; Heinz, Colburn, & Carney, 2001; Cohen, Furst, & Krips, 2004).

Hohn and Burkitt (2001) and Lowen and Teich (1996) derived an analytical description of neural processing assuming a stationary input. As most natural signals are not stationary, elucidating analytically the stochastic properties of each level in the neural pathway can lead to a more comprehensive analysis of the brain function.

In this letter, we analytically compute the probability density function of a set of spikes generated by a CD cell whose inputs are sets of spikes that behave as NHPP. Those derivations can be further used to predict binaural behavioral properties, for example, binaural perception (Krips & Furst, 2009; Krips, 2008).

## 2. Characterization of CD Cell Inputs

*t*,

_{i}*i*⩾ 1}. This series of action potentials behaves as a random point process with instantaneous rate λ(

*t*) and refractory period τ

_{r}. The probability of obtaining

*n*=

_{s}*n*spikes in the interval [0,

*T*) is equal to where

_{e}*n*is the number of generated spikes and Λ(

_{s}*T*) is the expected number of spikes in the interval [0,

_{e}*T*) and is given by The normalized instantaneous rate λ*(

_{e}*t*) is equal to the probability that a spike occurs at a given interval

*t*∈ [0,

*T*) given there are

_{e}*n*spikes in the interval [0,

*T*) (Snyder & Miller, 1991):

_{e}*Y*is an arbitrary event that occurred in the past, before the current interspike interval.

In the rest of the letter, we show that CD output exhibits NHPP behavior by equation 2.4. For simplicity and without loss of generality, in the rest of the letter, we choose *t _{k}* = 0.

## 3. An Excitatory-Inhibitory Cell Model

### 3.1. Simple Model with Two Inputs.

A simple excitatory-inhibitory (EI) cell has two asymmetric inputs: excitatory (E) and inhibitory (I). Their instantaneous rates are denoted by λ_{E} and λ_{I} for the excitatory and inhibitory inputs, respectively. Similarly, the refractory periods are denoted by τ^{(E)}_{r} and τ^{(I)}_{r}, respectively. An EI cell spikes whenever the excitatory input spikes and in the preceding Δ seconds (Δ>0), the inhibitory input does not spike. Therefore, if in the period *T* there are *n _{E}* and

*n*instances in the excitatory and inhibitory inputs, respectively, the EI cell will generate a spike at

_{I}*t*

^{(E)}

_{j}(1 ⩽

*j*⩽

*n*) if none of the inhibitory inputs

_{E}*t*

^{(I)}

_{i}(1 ⩽

*i*⩽

*n*) satisfies the condition 0 <

_{I}*t*

^{(E)}

_{j}−

*t*

^{(I)}

_{i}< Δ. To avoid the case that more than one excitatory spike can occur during Δ, we add to the definition of

*EI*the constraint Δ < τ

^{(E)}

_{r}.

*EI*cell output exceeds

*T*seconds. Equation 3.1 implies that is the event in which no spike was generated by

*EI*in

*T*seconds. This is denoted by According to the definition of an NHPP in equation 2.4, we will show in the following lemma that represents an NHPP, that is, its interspike intervals have an exponential distribution and they are statistically independent.

*If Δ < τ ^{(E)}_{r}, then when for , which is defined in equation 3.2. For any arbitrary event Y that occurred in the past, .*

*EI*includes the constraint Δ < τ

^{(E)}

_{r}, in an interval whose length is less than or equal to Δ, only one excitatory spike can occur. Let us further assume that there are maximum

*M*inhibitory and

*N*excitatory spikes in the interval [0,

*T*). The definition of in equation 3.2 yields the probability of , which is obtained by where, .

*t*

^{(E)}

_{p}−

*t*

^{(I)}

_{q}< Δ ∣

*n*=

_{E}*n*,

*n*=

_{I}*m*} that is included in equation 3.2 is obtained by Substituting equation 2.3 in equation 3.4 yields where λ

_{E}and λ

_{I}are equal to the expected number of spikes in an interval of length

*T*of the excitatory and inhibitory inputs, respectively, similar to the definition in equation 2.2.

*q*and p, the probability of given

*n*inhibitory spikes exactly inhibiting

*n*excitatory spikes is equal to . In case there are more inhibitory spikes than excitatory spikes (

*m*>

*n*), a total of (

*m*)

_{n}=

*m*!/(

*m*−

*n*)! different choices will satisfy equation 3.3, that is,

*M*≫ 1, . For

*n*>1, Γ

_{n}includes

*n*terms that are less than Γ

_{0}, that is, . It is easy to show that if for every term in the sum , which implies that if

*n*<

*M*+ 1 − Λ

_{I}then . Thus, for M ≫ 1, Substituting equation 3.9 in equation 3.8 yields where

### 3.2. Multiple Inhibitory EI Cell.

A multiple inhibitory EI cell has one excitatory input (E) whose instantaneous rate (IR) is λ_{E} and M independent inhibitory inputs {*I*_{1}, …, *I _{M}*} with instantaneous rates . The multiple inhibitory EI cell spikes whenever the excitatory input spikes, and in the preceding Δ seconds (0 < Δ < τ

^{(E)}

_{r}), none of the inhibitory inputs spikes. Therefore, the multiple inhibitory EI cell can be described as a string of M EI cells (see Figure 1). Each of the M EI cells receives a different inhibitory input. After the first simple EI cell in the string receives the original excitatory input (E), the second simple EI cell receives the output of the first simple EI cell as an excitatory input and so on. Since the output of a simple EI cell with independent inputs behaves as an NHPP, the output of the string of EI cells also behaves as an NHPP whose IR is given in equation 3.12.

*j*⩽

*M*− 1 in according to equation 3.13 yields It is clear from equation 3.14 that since the inhibitory inputs are independent, their order does not affect the resultant IR. Note that for

*M*= 1, equation 3.14 is equal to equation 3.12.

## 4. An Excitatory-Excitatory Cell Model

### 4.1. Simple EE Cell.

A simple EE cell has two independent symmetrical inputs (*E*_{1} and *E*_{2}). Each neural input is presented by a set of spikes that occur at instances and . The EE cell generates an output spike whenever both inputs spike within a time interval of less than Δ seconds (Δ ⩾ 0). EE generates a spike at *t*^{(EE)}_{K} if there exist and that satisfy condition and .

*t*is negligible, which implies that for any given

*i*and

*j*, . Thus, without losing the generality, we can assume that . Therefore, a simple EE cell output includes a set of spikes {

*t*,

^{EE}_{k}*k*>1} where Note that <Δ}. It is therefore possible to describe an EE cell as an OR combination of two cells that each can generate a spike that corresponds to the EE spike (

*t*

^{(EE)}

_{K}). We call each of those cells . They are sketched in Figure 2 as two-color circles. The white half receives the excitatory (

*E*) input and the gray half receives the noninhibitory () input. An output spike is generated if a noninhibitory () input was obtained at most Δ seconds before an excitatory spike. We chose the term

*noninhibitory*() input and cell, correspondingly, since its definition complements the definition of

*EI*cell (see equation 3.1). The difference between the two cells is that in the EI cell, the first spike (

*I*) inhibits the second spike (

*E*), while in the cell, the first spike () enables the second spike (

*E*).

*E*and are independent and are presented by a set of spikes: {

*t*

^{(E)}

_{i},

*i*⩾ 1} and . The output of the cell generates spikes at , where Both of the cells, which together build the EE cell, receive the original inputs

*E*

_{1}and

*E*

_{2}that act as

*E*or inputs alternatively. The cells' outputs are denoted as and , respectively. Thus, the possibility that may be excluded from the definition of an cell results in . The EE output includes the following series of spikes: , which have exactly the same incidences that were included in equation 4.1. In order to show that the EE cell output behaves as an NHPP, it is enough to prove that each cell output behaves as an NHPP.

*T*seconds. From equation 4.2, it implies that

As was indicated in equation 2.4, in order to prove that output behaves as an NHPP, we show in the following lemma that behaves as a descending exponential:

*For , which is defined in equation 4.3, and , then , where and is the refractory time of the type input. For any arbitrary event Y that occurred in the past, .*

*T*) as

*N*and for the

_{E}*E*and inputs, respectively. Note that for both inputs, the refractory time is positive, which yields a finite number of spikes in a finite interval. The actual number of spikes in the

*E*and inputs is denoted by

*n*and , where

_{E}*n*⩽

_{E}*N*< ∞ and . Based on the number of spikes entering the cell in the interval [0,

_{E}*T*), the event can be defined as a combination of independent events: and the probability of can be expressed as is given by From the definition of in equation 4.4, it follows that for

*n*>0 and , Since all the probabilities in equation 4.5b are NHPP and thus satisfy the following conditions, the implication is that equations 4.5a and 4.5b are identical, particularly . The probability of the event is obtained by where

_{E}*i*and

*j*, substituting its value in the probability for according to equation 4.6 yields Substituting equations 4.9 and 2.1 in equation 4.5 yields Since , it implies that

*N*≫ 1 and ; thus, and equation 4.10 becomes If , then , which yields Since , it follows that On the other hand, the maximum instantaneous rate is limited by the refractory time, which implies that It is therefore clear that if then necessarily and .

_{E}*E*and inputs, alternatively, EE cell output behaves as an NHPP if , and the resultant instantaneous rate is . From equation 4.8, it follows that which implies that

### 4.2. Multiple Input EE Cell.

A multiple input EE cell has *N* excitatory inputs {*E*_{1}, …, *E _{N}*} with instantaneous rates . It generates a spike whenever at least

*L*of its inputs spikes during an interval Δ.

The cell can be built as an OR combination of *L* strings of cells as depicted in Figure 3. Each string is built from cells, with each cell receiving a different noninhibitory input. The first cell in every string receives one of the *E _{i}* inputs as an excitatory input (

*E*) and another input

*E*as a noninhibitory input. The second cell in the string receives the output of the first cell as an excitatory input and

_{j}*E*as a noninhibitory input when

_{k}*k*≠

*i*,

*j*. The other cells in the string are built in a similar way. The cells can be seen in Figure 3.

*j*represents its position in the string. According to equation 4.16 and lemma 2, if , then the output of each string behaves as NHPP and the IRs along the string are given by Substituting for 1 ⩽

*j*⩽

*L*− 2 in according to equation 4.19 yields Since the inputs to the cell are independent, it is clear from equation 4.20 that the order of the cells in each string does not affect the resultant IR. Therefore,

*L*different strings contribute the IR of an cell output and are combined by an OR gate, as shown in Figure 3. The resultant IR is obtained by A multiple input

*EE*cell has

*N*independent inputs and generates a spike when at least

*L*of its inputs spike in an interval Δ. We now show that a multiple input

*EE*cell can be built from

*EE*

_{L}cells.

*E*

_{1}, …,

*E*} into subgroups, each containing two sets of inputs. One set will contain

_{N}*l*inputs to represent the inputs that generate a spike while a complementary set will include

*N*−

*l*inputs to represent the inputs that do not spike. Let us denote the firing inputs of the

*i*th group as and the nonfiring group as . For every choice of

*l*, there are different possibilities when

*L*⩽

*l*⩽

*N*. Therefore each of the possible subgroups includes all the

*EE*cell's

*N*inputs: For a given

*l*, there are exactly

_{i}*l*inputs that spike and

*N*−

*l*inputs that do not spike during the interval Δ. This situation can be described as a string of two cells. The first cell, of type , includes

*l*inputs, all of which should spike to generate an output spike. Its IR is derived by equation 4.21 and is denoted as to indicate the type of cell and the group to which it has been applied. The second cell is a multiple input

*EI*cell that receives the output of the as an excitatory input and

*N*−

*l*inhibitory inputs. Its IR is derived by equation 3.14.

## 5. General CD Cell

A general CD cell is defined as one with *N* excitatory inputs Ψ = {*E*_{1}, …, *E _{N}*} and

*M*inhibitory inputs, Ω = {

*I*

_{1}, …,

*I*}. This type of cell generates a spike if, during an interval of length Δ, there are at least

_{M}*P*more excitatory spikes than inhibitory spikes. A CD cell generates a spike in the following cases:

When exactly

*P*excitatory inputs fire and none of the inhibitory inputs fires during a time interval Δ. Such a case can be described by a multiple input*EI*cell that receives the set Ω as inhibitory inputs and the output of cell as an excitatory input (see Figure 4a). The resultant IR is derived from equation 3.14, which yields .- When
*P*+*K*excitatory inputs fire, and at most*K*of the*M*inhibitory inputs fire during the time interval Δ when 0 <*K*< min{*N*−*P*,*M*}. An*EI*cell represents this case (see Figure 4b). Its excitatory input is the output of cell that is applied on the set Ψ, and its inhibitory input is the output of that is applied on the inhibitory set Ω. The resultant IR is obtained by equation 3.12, which yields When

*K*>*M*and*P*+*K*⩽*N*, the CD cell will always generate a spike no matter how many inhibitory inputs are fired during the last Δ seconds. Such a case can be obtained only if*M*<*N*−*P*. The IR in this case will be the output of cell (see Figure 4c).

## 6. Sample Results

### 6.1. Decrease of Spontaneous Rate.

One of the properties of CD cells is a reduction in spontaneous rate (SR) when progressing across neural levels (Pfeiffer & Kiang, 1965; Tollin & Yin, 2002; Tollin, 2003; Ramachandran, Davis, & May, 1999).

*N*inputs and produces a spike when at least

*L*of them spike during an interval Δ. The IR of all of its inputs is constant and equal to λ

_{in}. The cell output IR is λ

_{out}, which is derived by equation 4.25, which yields

In Figure 5, λ_{out} is plotted as a function of *N*, for *L* = 2 and λ_{in} = 20 spikes/sec for different values of Δ. In order to maintain the condition for the Poisson process, we chose Δ · λ_{in} ⩽ 0.02. As can be seen in Figure 5, λ_{out} increases with *N* and Δ. However, as a function of *N*, it increases moderately. If some of the *N* inputs are injured and thus inactive, the output of the CD cell will remain unchanged if *L* ≪ *N*.

The increase in SR as a function of *N* can be moderated by a multilayered CD cell. The output of a low layer feeds the CD cells of the next layer.

A network of type cells was designed. To maintain the independence of the inputs to each cell, the number of cells in every layer (*g*) was equal to *N*^{G−g}, when *G* is the maximum number of layers. λ_{out} of each layer was derived for λ_{in} = 100 spikes/sec, Δ = 0.5 mSec, and *L* = 2 for different values of *N* and is plotted in Figure 6. It is clear from Figure 6 that both robustness and decrease in SR can be achieved. Increasing *N* causes an increase in SR but improves robustness. However, by increasing the number of layers, a significant decrease in SR can be gained.

### 6.2. SNR Improvement for Sinusoidal Inputs.

*A*>1, to guarantee λ

_{in}>0. The initial SNR is thus The IR of a simple EE cell was derived when both inputs have λ

_{in}, as indicated in equation 6.2, and yielded For Δ · ω ≪ 1, equation 6.4 can be approximated as which yields

The improvement in SNR as a function of the input SNR is shown in Figure 7a. The maximum SNR improvement of 6 dB was achieved for a relatively low-input SNR. The effect of Δ on the SNR improvement is shown in Figure 7b for different values of *A*. The SNR improvement vanishes with the increase of Δ, as can be expected from an efficient correlation processing.

_{n}), ω

_{n}≠ ω

_{s}; for example, the IR can be In order to reduce both components, the DC and ω

_{n}, relative to the desired component, ω

_{s}, a two-layer network as presented in Figure 8 can be applied. The first layer includes two EI cells, each receiving excitatory and inhibitory inputs. The IR of the excitatory input is obtained by where λ

_{1}was defined in equation 6.7. The IR of the inhibitory input is given by The normalized IRs, λ*

_{E}and λ*

_{EI}are plotted in Figure 9 in both time (panel a) and frequency (panel b) domains. The decrease of the ω

_{n}component at the output of the first layer (λ*

_{EI}) is clearly shown. When a second layer of EE is applied, the ω

_{s}component is significantly increased, as indicated in Figure 9b by λ*

_{out}. In general, EE cells increase common elements, whereas EI cells remove them.

### 6.3. LSO Cell's Response to Interaural Level Difference.

Interaural level difference (ILD) is one of the most important cues for localizing high-frequency sounds (Irvine, 1992; Blauert, 1997). In mammals, ILD-sensitive neurons are found at almost every synaptic level from brainstem to cortex. The first nucleus is the LSO (Boudreau & Tsuchitani, 1968; Caird & Klinke, 1983; Park, Klug, Holinstat, & Grothe, 2004). LSO cells receive excitatory inputs from the ipsilateral cochlear nucleus and inhibitory inputs from the contralateral medial nucleus of the trapezoid body, which is driven by the contralateral cochlear nucleus.

*M*is the number of inhibitory inputs and Δ is the coincidence interval. is plotted as a function of ILD in Figure 10. The derived mean rate was compared to experimental data (Tollin & Yin, 2002). The data points indicated in Figure 10 were obtained from Figure 1E in Tollin and Yin (2002). The data were collected from the LSO unit of an adult cat. The stimulus was 16 kHz tone, and the excitatory ear was held fixed at 30 dB SPL (

*A*= 30). The level to the contralateral ear (

_{ipsi}*A*) was varied from 5 to 55 dB SPL. In order to predict the data, we used equation 6.10 with spikes/sec, and

_{contra}In order to fit equation 6.10 to the experimental data, we chose m sec. The data were fitted with . For every Δ, we calculated the correspondent number of inhibitory inputs (*M*_{Δ}) that yielded the minimum mean square error. All pairs (Δ, *M*_{Δ}) yielded a similar fit, as can be seen by the solid lines in Figure 10. The different lines merged almost to a single line. The dependence of *M*_{Δ} on Δ is shown in Figure 11, which reveals an increase of *M*_{Δ} with the decrease of Δ. Some studies indicate about 10 inhibitory inputs to LSO cells (Sanes, 1990; Shi & Horiuchi, 2004). Other studies indicate a larger number of inputs (Reed & Blum, 1990). A more extensive experimental study is required to resolve this issue.

## 7. Discussion

In this letter, we presented the stochastic properties of CD cells with both excitatory and inhibitory inputs, which we derived based on the initial assumption that all the inputs to the CD cell are independent and behave as NHPP. We proved analytically that both EE and EI outputs behave as NHPP. EI output requires that the coincidence interval (Δ) be shorter than the minimum refractory period of all its inputs, while Δ of EE is significantly smaller than the refractory period. This result means that the decision mechanism for generating a spike in a CD cell is terminated before any of its inputs generate a successive spike. In other words, the processing rate of a CD cell is faster than the incoming information rate, reflecting a robust, efficient, and reliable mechanism.

For the inhibitory inputs, the integration time must be smaller than the input information rate, while for the excitatory inputs, the integration time should be much smaller. We have shown that in general, EE cells increase common elements, whereas EI cells remove them. It seems that a shorter period of time is devoted to discovering and strengthening common elements, while a longer period is allowed for removing them.

It is quite clear that EE cells with their multiple inputs guarantee robustness, particularly in the case of lesions. Indeed, the system can perform almost normally when some of its parts are not functioning. While there will be some deterioration in the system's performance in the presence of serious defects, some functionality will remain.

Studies that measured Δ in different brain structures report values of the order of 10 microseconds and up to 100 microseconds in the lower parts of the auditory pathway (for excitatory inputs: Skottun, Shackleton, Arnott, & Palmer, 2001; Wagner, Brill, Kempter, & Carr, 2005; Agmon-Snir et al., 1998; Heinz, Colburn, & Carney, 2001; for inhibitory inputs: Brand, Behrend, Marquardt, McAlpine, & Grothe, 2002; Grothe, 2003; Siveke, Pecka, Seidl, Baudoux, & Grothe, 2006). The refractory period in those systems is on the order of 300 microseconds up to 1.3 milliseconds (Miller, Abbas, & Robinson, 2001; Bruce et al., 1999; Li & Young, 1993; Brown, 1994). At higher levels of the brain, the coincidence interval Δ is much higher and might reach milliseconds (Larkum, Zhu, & Sakmann, 1999). The refractory period can increase to the order of hundreds of milliseconds or even seconds (Larkum & Zhu, 2002; Sanchez-Vives & McCormick, 2000). Therefore, the requirement that Δ is significantly smaller than the refractory period of the coincidence inputs is valid for different brain structures.

The statistical properties of CD cell output were obtained by analytic calculation of the probability density function (pdf) of its spike train. The CD output was found to behave as an NHPP whose only parameter is the instantaneous rate (IR). The CD output's IR depends on only its inputs' IRs. The resulting closed form for the CD output IR implies that for any given feedforward neural system with CD cells, it is possible to calculate the pdf of the output of the entire system.

In this letter, we demonstrated this capability of the EE and EI cells with three simple examples: (1) a decrease in the spontaneous rate in the higher levels of the brain, (2) an improvement of the signal-to-noise ratio in EE cell output with sinusoidal (harmonic) inputs, and (3) a discharge rate change of EI cell as a function of interaural difference (ILD). Other properties of CD cells and their correspondence to binaural psychophysical performances were presented in detail in Krips and Furst (2009). In particular, it was shown that interaural time delay (ITD) is primarily estimated by EE cells and the ipsilateral auditory input exhibits a phase delay between 40 and 65 degrees as physiological data suggest (Yin & Chan, 1990; Palmer et al., 2002; McAlpine & Grothe, 2003; Hancock & Delgutte, 2004; Joris & Yin, 2007). ILD, on the other hand, is most likely estimated by EI cells, as was shown by physiological data (Tollin & Yin, 2002; Sanes, 1990; Reed & Blum, 1990; Shi & Horiuchi, 2004; Glendenning et al., 1985; Cant, 1991; Warr, 1972; Boudreau & Tsuchitani, 1968).

CD cells are frequently found in different neural structures (Abeles, 1991). They can receive inputs from the same or different modalities. For example, CD cells have been identified in different levels of the auditory system and recognized as an essential part of the localization mechanisms (Jeffress, 1948; Joris et al., 1998; Joris & Yin, 2007; Grothe, 2003; Kandler, 2004; McAlpine, 2005; Tollin & Yin, 2002).

The tools developed in this letter can serve in analyzing CD neural cell networks, which can lead to a better understanding of how the brain functions. We demonstrated the functionality of signal separation for EI and the integration of EE and EI into a single network. As well as presenting improved signal enhancement abilities, the network introduces the methodology of integrating both EE and EI components into a single cell since both result in NHPP spike trains.

The mathematical derivation in this letter was limited to a feedforward system whose inputs are independent and equally weighted. Most neural systems include both feedback (i.e., dependent inputs) and unequal weighted inputs (Kandel et al., 2000). It is possible to confront this discrepancy by assuming that for feedback paths that are long enough (e.g., periphery versus higher brain levels), the input that arrives from higher levels of the brain is independent of other inputs that arrive from lower parts in the brain. In order to account for the different weighted inputs to CD cells, it is possible to generalize from the derivation shown in this letter by referring different coincidence intervals to each of the different inputs. In this way, inputs with longer coincidence intervals can be considered stronger than inputs with shorter intervals. The condition for obtaining a NHPP in the CD output can then be expressed separately for each input (i.e., its coincidence interval should be smaller than its refractory period of the relevant input). However, the general result holds: the output behaves as an NHPP.

## Acknowledgments

We thank Eli Merzbach and Yair Shaki for their inspiring discussions and comments.