Conventional artificial neural networks (ANNs) model synaptic transmissions as the sole source of neural signaling. Yet, research in cellular neurophysiology has concluded that nonsynaptic communication exists between neurons [1] [2] [3]. One method of nonsynaptic signaling is volume transmission (VT) [4], as shown in Figure 1.1. A substance is defined to be a VT signal if: 1) it is released by a cell in a regulated fashion; 2) diffuses at a relevant concentration in the extracellular fluid for a distance larger than the synaptic cleft; 3) is able to activate selective receptive molecules in a number of target cells; and 4) triggers a physiological response in the target cells [5]. Agnati further proposes that extracellular pathways diffusely connect neurons [4] [5].

In this paper, concepts related to VT are applied to a biologically motivated artificial neural network (ANN). Using the anatomy of housefly's (Musca domestica) first optic ganglion, the lamina ganglionaris , a simplified ANN, which uses extracellular diffusion (ECF) to model volume transmission, was developed. This work assumes that the extracellular space (ECS) surrounding a collection of neurons maintains an extracellular potential (ECP) resulting from currents produced by the axons of these neurons. The exploration into the functionality of ECD was investigated using a computer model of the ANN. This paper presents and discusses simulation results investigating the pulse response of the ANN.

The next section gives an overview of those neurons in the fly's visual processing which are relevant to the ANN. Section 3 presents the model and a discussion of the derivation of the system of differential equations, which are used to describe the evolution of the neural potentials is provides. In Section 4, testing procedures and results are given followed by a discussion of results in Section 5.

A brief background of the fly's visual processing system, which was the motivation for the ANN, is presented. A more detailed review of this system can be found in [6]. The optic lobe of the housefly is configured into four collections of neural processes (neuropiles) organized into multiple sets of retinotopical pathways [7] [8], as simplified in Figure 2.1. Each neuropile is a functionally specialized region that processes certain components of the fly's visual processing [8]. The visual processing occurring in the first neuropile, the lamina, coalesces the visual stimuli, enabling the visual system to transmit as much information as possible through channels (neurons) of limited bandwidth [9]. This research focuses on the visual processing within the lamina neuropile.

The retina of the housefly is composed of hexagonally arranged eye structures called ommatidia (little eyes) [10]. Each ommatidium is a columnar structure containing eight photoreceptor (light receiving) neurons commonly labeled R1-R8. Cross-sections of an ommatidium reveal a rhomboidal pattern with six large photoreceptors, R1-R6, surrounding a pair of smaller central ones, R7-R8 [11]. R1-R6 send their axons to different synaptic cartridges within the lamina, crowning a pair of large monopolar cells (LMCs) with which they synapse [12] [13]. Within the lamina, presynaptic to the LMCs, gap junctions electrically interconnect the neighboring photoreceptor axons of R1-R6 [14]. The axons of R7 and R8 bypass the LMCs within each cartridge and terminate in the second neuropile, the medulla [15].

Each synaptic cartridge within the lamina receives axons from eight photoreceptors (R1-R8), each from a different ommatidium, possessing similar visual axes [16]. This is illustrated in Figure 2.2. In sharing similar visual axes, these groups of photoreceptors sample similar regions in visual space. This convergence of information from the visual environment into one synaptic cartridge is known as the neural superposition principle [12] [16].

The lamina lies between the retina and the outer chiasma in the optic lobe of the fly [8]. A basement membrane with a high electrical resistance, called the outer-plexiform layer, separates the retina and lamina [17]. The lamina is composed of synaptic cartridges or neuro-ommatidia with the number of neuro-ommatidia equaling the number of ommatidia in the retina [8]. As with the pattern of the ommatidia in the retina, the synaptic cartridges within the lamina are arranged hexagonally. A high electrical resistance separates each synaptic cartridge and as a result, encases cartridge potential [17].

Within each synaptic cartridge, the axons of R7 and R8 bypass lamina processing. In contrast, the axons of R1-R6 surround a pair of LMCs, L1 and L2, which project centrally and synapse in the medulla [13] [18]. From the size of their axons, L1 and L2 seem to be the most important relay neurons of the fly's lamina [8]. LMC processing inverts and amplifies the photoreceptor input signals [19].

One premise to the functionality of ECD in neural processing is that the potential acting at the photoreceptor-LMC synapse is the difference between the photoreceptor axon terminal potential and the ECP [20] [21]. The depolarization of the ECP, due to the flow of light-induced currents from the retina to the lamina, will reduce the potential at the photoreceptor axon terminal, affecting the LMC's input [20]. Another proposal to the role of ECD is that the ECP diffuses between adjacent cartridges giving rise to a source of nonsynaptic transmissions between the synaptic cartridges [22].

Figure 2.2:   Neuro-ommatidial converging visual pattern.
Each synaptic cartridge (neuro-ommatidium) in the lamina receives axons from eight photoreceptors (R1-R8), each from a different ommatidium, possessing similar visual axes. In sharing similar visual axes, these groups of photoreceptors sample similar regions in visual space. This convergence of information from the visual environment into one synaptic cartridge is known as the neural superposition principle.

A simplified model of the fly's lamina processing, which can be seen in Figure 3.1, served as the basis for an ANN presented in this work. The ANN contains multiple artificial lamina cartridges (ALCs) arranged in a two-dimensional hexagonal grid, as shown in Figure 3.2. Each ALC receives input from a set of seven artificial photoreceptors (ARs), AR1-AR7, possessing similar visual fields. Because R7 and R8 form a single visual axis, these axons of have been combined into one artificial photoreceptor, AR7.

Unlike the fly's asymmetric arrangement neuro-ommatidial visual field sampling, for simplicity, the AR's visual field sampling is symmetric. Six peripheral ARs receive stimulus from visual fields surrounding the central field of view of AR7. Stimulus-induced potential is localized in that only the ARs receive external visual information. AR1-AR6 provide input into a pair of artificial large monopolar cells (ALMCs), AL1, and AL2. These ARs are connected to neighboring ARs to form a ring structure. The longer axon of AR7 has no connections and bypasses LMC processing.

The ANN uses the artificial extracellular space (AECS) as a medium for ECD. Hence, each ALC contains an AECS, which acts as a diffusive intermediary for relaying the activities between adjacent ALCs. Diffusive signaling also occurs between the AECS and its respective ARs. Thus, AR potentials are diffused into their respective AECS, and the resulting AECS potential in turn, diffuses in into adjacent ALCs. Consequently, AR and AECS activities are coupled, as is the activity between adjacent ALCs.

The AR inputs into each ALMC are modeled as chemical synapses. Accordingly, as AR potential is transferred to each ALMC, it must cross the AECS. In applying ECD to synaptic transmissions, differences between AR potentials and the AECS potential are used as inputs into each ALMC, rather than just the AR potential. ALMC output provides the output for the ANN, which are open for comparison with the unprocessed stimulus of AR7. This section continues with a brief discussion on the derivation of stimulus, activation, and output system of equations.

Figure 3.2:   Hexagonal arrangement of artificial lamina cartridge (ALC) array.
Shown is a 4 × 4 array of ALCs arranged in two-dimensional hexagonal coordinate system. Each ALC contains seven ARs, AR1-AR7, two ALMCs, AL1 and AL2. These artificial neurons are surrounded by an AECS. Six ARs, AR1-AR6, provide input into AL1, and AL2. AR1-AR6 are laterally connected to neighboring ARs. The longer axon of AR7 has no connections and bypasses ALC processing. The ANN considers the AECS a medium for nonsynaptic interactions amongst the network of ARs and ALCs.



3.1    Model Processing

Model processing is subdivided into three levels: stimulus, activation, and output processing. Stimulus processing calculates the amount of light stimulus within each AR's visual field. Activation processing updates AR and AECS potentials. Output processing calculates the outputs of the ALMCs, AL1, and AL2. It is acknowledged that the calculations provided are greatly simplified compared to the actual processing performed within the fly's lamina. However, it is not the intent of this research to provide a precise neurophysiological model of the fly's early visual processing but rather to extract the computational principles found within the lamina. A synopsis of the ANN processing is next provided. Details on the derivation of the model processing can be found in [6] The units for parameters and variables discussed in this section are summarized in Table 3.1.

In modeling the stimulus value, I⁺, the complexity of the ommatidium to neuro-ommatidium mapping was bypassed by directly considering the visual fields ultimately associated with each AR ensemble. To account for the focal point changes due to varying angles of light incident to the ommatidia, AR visual fields were convolved with a two-dimensional Gaussian weighting function. Thus, for the i^th AR in the k^th ALC, AR _ki, the calculated stimulus input value, I⁺_ki (x,y), is the average of its sum of Gaussian weighted visual field stimulus values.

Activation processing updates the AR and AECS potentials where a_ki represents the potential for the i^th AR in the k^th ALC and z_k, represents the AECS potential within the k^th ALC. Electrical circuits were used to model these processes and current flow analysis was applied to derive the differential equations necessary to calculate the updates to the AR and AECS potentials, da_ki/dt and dz_k/dt. The reader is referred to [6] for the diagrams of the electrical circuit models of ANN processes.

Membrane leakage, stimulus, ARs connected by gap junctions, and ECD between an AR and its respective AECS are accounted for in the equivalent electrical circuit of AR processing. Given an AR membrane leakage conductance, A , a stimulus-induced conductance, I⁺, a gap junction conductance, D, and an AR/AECS channel conductance, E, the generalized equation used to determine updates to the AR potential, a_ki/dt is

The electrical circuit modeling AECS processing accounts for the ECD occurring between adjacent ALCs and contributing AR axons. Thus, extracellular potential is affected by the potential of its ARs, as well as by the AECS potential from its adjacent ALCs. Given the six potentials from each AR, a_ki, the AECS potential, z_k, an AR/AECS channel conductance of E', and an AECS/AECS channel conductance of L, the generalized equation used to calculate the updates to AECS potential, dz_k/dt, is

The differences between the AR AR/AECS channel conductance, E and the AECS AR/AECS channel conductance, E' , are to account for AR terminal volume and AECS volume differences. Specifically, for the fly, the estimated volume of the ECS is 9.09% of the AR terminal volumes [23] ALMC processing evolves the ALMC potential where a_Li, represents the potential for the i^th ALMC. An electrical circuit was used to model ALMC processing and current flow analysis was applied to derive the equation, which determined changes to the ALMC potential. To account for the effects of ECD, the actual input into each ALMC are the AR-AECS potential differences. In addition, ALMC processing considers ALMC membrane leakage. Given an ALMC membrane leakage conductance, F, and letting the variable G_i signify the gain for the ith AR-AECS input, the generalized equation used to calculate the changes to the ALMC potential, da_Li/dt, is

It is noted that G_i can also be viewed as the synaptic weighting of the i^th AR-AECS potential input. It is noted that AR-AECS is used to denote input resulting from AR and AECS potential differences. This differs from the notation, AR/AECS, which is used to denote diffusive channels connecting AR and AECS activities. These resulting set of differential equations used evolve ANN processes are nonlinear (bilinear) and coupled, making an analytical solution difficult to derive. Therefore, to investigate the response of this model to various inputs, a computer model was developed. Simulations results of this computer model are next presented.

A, D, E, E', F, L, I+	Siemans/Farad
B, C	Volts/Farad
G	1/Farhad
aki, zk, aLi	Volts
daki/dt, dzk/dt, daLi/dt	Volts/second

Table 3.1: Units of Measurement.
Shown are the units of measurement for the parameters, AR, AECS, and ALMC potentials, and updates to these potentials discussed in this paper. A detailed derivation of these units can be found in [6].

A computer simulation of the ANN was developed and numerical experiments were conducted. The computer model relies on standard numerical integration methods to approximate the solutions. The experiments presented in this paper studied the pulse response of the ANN model. For this, all ARs were initially stimulated with a background stimulation of 1.0, as shown in Figure 4.1. After two seconds, a pulse signal of two seconds and amplitude of 5, was applied to all ARs within a selected the ALC, while the ARs within the remaining ALCs received the background stimulation.

Pulse response simulations employed an 8 × 8 array of ALCs. Model parameters held constant throughout testing were A = 0.7, B = 1.0, C = 1.0, D = 0.25, E = 0.1, F = 3.0, G = 10.0, L = 30.0. Most parameter values were selected to closely relate to the parameters in Moya's model [23]. Moya's studies concluded that the optimal AR membrane leakage conductance was close to 0.7 and gap junction conductance was 0.22 [23]. The parameters B and C were selected to limit the AR output to be less than 1.0. The remaining parameters were empirically selected using the results of initial experiments. During testing, AR and AECS activation values were initialized to 0.5, which is one-half the maximum AR value of B/C. ALMC activation values were initially set to zero so that the initial ALMC responses were entirely dependent upon the AR and AECS potentials. Figure 4.2 provides a mapping of the responses provided in pulse response investigations. Figure 4.3 displays the pulse response of one stimulated AR, AR0, the response of the same AR in an adjacent ALC, AR1, and an AR in a distant ALC, AR2. Figure 4.4 shows the pulse responses for the AECS surrounding stimulated ARs, AECS0, the response of the AECS in an adjacent ALC, AECS1, and the AECS of a distant ALC, AECS2. Figure 4.5 displays the AR-AECS potential differences used as inputs into the ALMCs. Shown in this figure is the input resulting from the stimulated ARs, IL0, and the input applied to the ALMC in an adjacent ALC, IL1, and a distant ALC, AL2. Figure 4.6 provides the pulse responses for the ALMCs receiving input from stimulated ARs, ALMC0, and the response of an ALMC in an adjacent ALC, ALMC1, and a distant ALC, ALMC2. The simulation results presented in this section are next discussed.

Figure 4.2:   Pulse response orientation map.
Shown is a mapping ANN components from which response measurements were taken during pulse response simulations. Measurements taken include the AR, AECS and ALMC responses from the ALC receiving stimulation (AR0, AECS0, ALMC0). Measurements also include responses of the same components in an adjacent ALC (AR1, AECS1, and ALMC1) and a distant ALC (AR2, AECS2, and ALMC2).

Figure 4.3:   Comparison of AR pulse responses.
Results of pulse response simulations. ANN parameters held constant throughout testing were A = 0.7, B = 1.0, C = 1.0, D = 0.25, E = 0.1, F = 3.0, G = 10.0, and L = 30.0. Shown are the responses of the stimulated AR, AR0, the AR response in an adjacent ALC, AR1.

Figure 4.4:   Comparison of AECS pulse responses.
Results of pulse response simulations. ANN parameters held constant throughout testing were A = 0.7, B = 1.0, C = 1.0, D = 0.25, E = 0.1, F = 3.0, G = 10.0, and L = 30.0. Shown are the responses of the AECS surrounding stimulated ARs, AECS0, the AECS response in an adjacent ALC, AECS1, and a distant ALC, AECS2.

Figure 4.5:   Comparison of ALMC input pulse responses.
Results of pulse response simulations. ANN parameters held constant throughout testing were A = 0.7, B = 1.0, C = 1.0, D = 0.25, E = 0.1, F = 3.0, G = 10.0, and L = 30.0. Shown are the inputs applied to the stimulated ALMC, IL0, the input applied to ALMCs in an adjacent ALMC, IL2, and the input into ALMCs into a distant ALC. IL2.

Figure 4.6:   Comparison of ALMC pulse responses.
Results of pulse response simulations. ANN parameters held constant throughout testing were A = 0.7, B = 1.0, C = 1.0, D = 0.25, E = 0.1, F = 3.0, G = 10.0, and L = 30.0. ANN parameters Shown are responses of the stimulated ALMC, ALMC0, the response of an ALMC in an adjacent ALC, ALMC1, and a distant ALC. ALMC2.

The AR responses presented in Figure 4.3 demonstrate that AR potential is a function of the stimulus strength and consequently the greater the stimulus, the greater the potential. For sustained stimuli, the AR potential converges to an asymptotic value. The comparison of the pulse response for the stimulated AR, AR0, to the same AR in an adjacent ALC, AR1, and a distant ALC, AR2, show that AR1 and AR2 are unaffected. This implies that diffusive mechanisms have little or no effect on the ARs in adjacent ALCs.

The AECS responses shown in Figure 4.4 reveal that AECS potential is correlated to AR potential. Thus, an increase in the AR potential results in an increase in AECS potential. More significant are the simulation results for AECS1. These results show that, although the ARs contributing to AECS1 are not stimulated, AECS1 is still stimulated. The AECS response in a distant ALC, AECS2, is minimal. These results support the premise that extracellular potential resulting from ECD produces a nonsynaptic pathway connecting synaptic cartridges.

The comparison of the ALMC inputs to their respective provided in Figure 4.5 to their respective AR and AECS responses show that the ALMC input signal is an amplification of contributing AR-AECS potential differences. Signal attenuation is particularly noticeable when comparing IL1 to AR1 and AECS1. This demonstrates that ECD can influence neurons in adjacent cartridges. The comparison of ALMC inputs also illustrates the on-center/off-surround characteristic resulting from AR-AECS potential differences. Because this not a characteristic of AR and AECS responses, this characteristic is attributed to the differencing of extracellular and synaptic potentials. Results also demonstrate that the potential consequential to diffusive mechanisms reduces the background potential present in the both signals. This is demonstrated by the elimination of the background potential in the ALMC inputs.

ALMC responses shown in Figure 4.6 illustrate that ALMC processing inverts the input signals received. The comparison of ALMC0 to ALMC1 and ALMC2 shows the off-center/on-surround characteristic of ALMC processing. This means that in response to an increase in stimulus, ALMCs are self-inhibiting and excite ALMCs in adjacent ALCs. As with the AECS1 and IL1, ALMC1 is stimulated even though its respective ARs were no directly stimulated. The response of the ALMC2 reveals that electrical potential from the stimulated ARs is relayed to ALMCs in distant ALCs.


6     Conclusion

Beginning with the concept that ECD can contribute electrical neural processing, this work has yielded several significant results. This includes a biologically inspired ANN of the fly's early visual processing system. The ANN exemplifies the concept that the extracellular within a synaptic cartridge maintains a potential resulting from currents produced by photoreceptor axons. The results presented support the premise that extracellular potential resulting from ECD produces a nonsynaptic pathway o connecting synaptic cartridges. Consequently, the ECP in one synaptic cartridge can affect the ECP in adjacent cartridges. Results will also show that the potential consequential to diffusive mechanisms between the presynaptic (input) potential and ECP reduces the background potential present in the both signals. Changes are of sufficient magnitude to substantially reduce the driving potential at synapses in adjacent cartridges resulting in an on-center/off-surround response to a spot of light. These conclusions support the hypothesis that ECD is has a significant functionally in electrical neural processing.


7     Acknowledgements

This report was supported in part by the IEEE Neural Network Summer Research Grant Program, was submitted and accepted for publication at the International Joint Conference on Neural Networks, 2001.


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