Modeling Nonsynaptic Communication Between Neurons in the Lamina Ganglionaris of Musca domestica Karen G. Haines†, John A. Moya‡, Thomas P. Caudell† † Dept. of EECE, University of New Mexico, Albuquerque, NM 87131 ‡ Dept. of EE, New Mexico Tech, Socorro, NM 87801
[email protected],
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[email protected] Abstract Using anatomy of Musca domestica’s (the housefly's) first optic ganglion, the lamina ganglionaris, a biologically inspired artificial neural network (ANN) is presented. This paper focuses on communicational phenomena and activity between specific neurons in the lamina and models nonsynaptic communication between neurons. Test results demonstrate that nonsynaptic communication between neurons can achieve results similar to synaptic connections in early vision systems.
1 Introduction Most artificial neural networks (ANNs) model synaptic transmissions as the sole source of neural communication. Yet research in cellular neurophysiology has concluded that nonsynaptic communication exists between neurons. For example, ephaptic interactions presume electrical fields in extracellular space generated by active neurons induce current flow in adjacent neurons [1,2] and volume transmission proposes that unstructured extracellular fluid pathways diffusely connect neurons [3,4]. Although nonsynaptic modulation of neuronal activity is surmised to influence neural communication, its functional role remains to be defined. In the following, concepts related to nonsynaptic communication are extended to an artificial neural network (ANN) to investigate a source of lateral communication between neurons. This research focuses on activity between neurons of the first optic ganglion, the lamina ganglionaris, in the optic lobe of the housefly, Musca domestica.
2 Background The optic lobe of the housefly can be delineated into four collections of neural processes (neuropiles) organized into multiple sets of retinotopical pathways and each neuropile subserves certain components of the fly's visual processing [5]. The first stages of visual processing occurring in the first neuropile, the lamina, coalesces the visual stimuli [6,7] enabling the visual system to transmit as much
information as possible through channels (neurons) of limited bandwidth [6,8,9]. The compound eye of the housefly is composed of hexagonally arranged eye structures called ommatidia (little eyes). Each ommatidium is a columnar structure containing eight retinula or photoreceptor (light receiving) neurons commonly labeled R1-R8. Cross sections of ommatidia reveal an asymmetric rhomboidal pattern with six large peripheral photoreceptors, R1-R6, surrounding a pair of smaller central ones, R7-R8 [10]. Each photoreceptor cell is associated with a rhabdomere and each rhabdomere within an ommatidium has a different visual axis. The rhabdomeres of R7 and R8 are arranged one above the other and form a single visual axis [11]. The peripheral photoreceptor axons, R1-R6, send their axons to different synaptic cartridges within the first optic neuropile, the lamina ganglionaris. Within a synaptic cartridge, gap junctions electrically interconnect the peripheral photoreceptor axons and allow the activity of one photoreceptor to flow to its neighboring photoreceptors [10]. The photoreceptor axons of R7 and R8, pass through the lamina and terminate in the second optic ganglion, the medulla [12,13]. The lamina is composed of multiple hexagonally arranged synaptic cartridges or neuro-ommatidia with the number of neuro-ommatidia equaling the number of ommatidia in the retina [5]. Each cartridge receives the axons from a set of photoreceptors, each from different ommatidia possessing similar visual fields as shown in Figure 1. The photoreceptor axons, R1-R6, crown and provide input into a set of large monopolar cells (LMCs) [11,14]. The LMCs transform the photoreceptor signals and transmit the results to the ensuing neuropile, the medulla. In sharing similar visual axes, the photoreceptor axon input ensemble sample similar regions in space. In this way, six times as many light quanta are used to measure brightness in the overlapping fields of view. This convergence of
photoreceptor, AR7. For simplicity, the input arrangement of the ARs is symmetric with six peripheral ARs, AR1AR6, surrounding a central AR, AR7. Each AR maintains a unique activity state that is sensitive to the amount of stimulus present in its respective visual field. Within the ALC, the six peripheral ARs, AR1-AR6, surround and crown a set of artificial large monopolar cells (ALMCs), AL1 and AL2. These ARs are laterally connected to neighboring ARs. The longer axon of AR7 has no connections and bypasses ALC processing. Outputs of the ANN include the unprocessed stimulus of AR7 and the outputs of AL1 and AL2.
Figure 1. Field of view associated with a photoreceptor within Musca’s ommatidia and its respective mapping into the neuro-ommatidia with the lamina ganglionaris of the optic lobe. information into one synaptic cartridge, known as the neural superposition principle [14,15], enhances the intensity of the retinal image without compromising spatial acuity [15]. Cartridges are readily resolvable because they are surrounded and separated by epithelial glial cells (EGCs) which isolate cartridge activity [16]. It is assumed that the EGCs form an extracellular space (ECS) or medium through which activities between adjacent cartridges diffusely interact. It is also presumed that ECS activity diffuses across the membranes of its contributing photoreceptor axons. These conjectures give rise to a source of nonsynaptic lateral interactions within a network of neurons.
3 The Model Architecture A simplified architecture of the fly's photoreceptor axon and cartridge processing, as shown in Figure 2, is applied to an artificial neural network (ANN). The ANN contains multiple artificial lamina cartridges (ALCs) arranged in a two-dimensional hexagonal coordinate system. Each ALC receives the axons of seven artificial photoreceptors (ARs), labeled AR1-AR7, possessing similar visual fields. Since the long photoreceptors form a single visual axis, the axons of R7 and R8 have been combined into one artificial
The ANN assigns to each ALC an activity state representing an artificial extracellular space (AECS). The ANN considers the AECS a medium of nonsynaptic interaction amongst the network of ARs and ALMCs. Specifically, the AECS around the ARs is modeled as a diffusive intermediary for relaying the activities between the ALMC's AECS and its contributing ARs. AECS activities are also diffused between adjacent ALMCs. Consequently AR and AECS activities are coupled, as is the activity between neighboring ALMCs.
4 Model Processing The ANN's processing is subdivided into three levels: stimulus, activation, and output processing. Stimulus processing calculates the amount of light stimulus within an AR's visual field. Activation processing updates AR and AECS activity states. Output processing calculates the outputs of the ALMCs, AL1 and AL2. These outputs are open for comparison with the unprocessed stimulus of AR7. For this paper, only activation processing is discussed. 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 cellular neurophysiological model of activities occurring in the fly's synaptic cartridge but rather to extract the computational principles found within the lamina.
4.1 Activation Processing Signaling between neurons relies on changes in the electrical potential difference across cell membranes [17]. Changes to the membrane potential are produced by changes in ionic current through channels across the membrane which drive the potential away from its resting value [18]. Neuroscience has established that mathematical models derived from electrical circuits can be used to describe this electrical behavior of membranes. Typically, the dielectric property of membranes is represented as a capacitor and the permeation of ions through a channel or the channel conductance is represented as a resistor. An
circuit and using Kirchhoff's Current Law, which states that the sum of currents entering a node equals the sum of the currents leaving the node, yields Cadaki/dt = - ileak - istimulus + igap - iAR/AECS
(1)
where Ca is the AR capacitance, ileak is the membrane leakage current, istimulus is the current due to the stimulus received, igap is the gap junction current, and iAR/AECS is the current through AR/AECS channels.
Figure 2. Simplified model of the fly's photoreceptor axon and synaptic cartridge processing. equivalent electric circuit model is applied to the ANN and current flow analysis is used to derive the equations necessary for activation processing. Activation processing updates two activities: AR and AECS activities. ANN processing assumes these represent electrical potentials or voltages. In particular, the current state of the AR activity, aki, models the potential for the ith AR in the kth ALC. Similarly, the AECS Activity State, zk, represents the extracellular potential within the kth ALC. In examining the model depicted in Figure 2, it was determined that there are five channels through which current can flow. These include (1) membrane leakage channels, (2) channels sensitive to stimulus, (3) gap junctions, (4) AR/AECS and (5) AECS/AECS diffusive interactions that are viewed as channels. Electrical circuits represent these channels and current flow analysis was applied to determine the equations describing changes to the AR and AECS potentials, daki/dt and dzk/dt. Analysis used the convention that current flowing into a cell is positive and current flowing out of a cell is negative.
Membrane leakage channels are modeled as continuously opened channels with a finite and constant resistance, 1/A. For the ith AR in the kth ALC, the driving force of the leakage current, (aki - VR - VRP), is the electrical gradient produced by the potential difference between the AR potential, aki, and the potential that is outside the ALC, VR. In addition, there exists a force or resting potential, VRP, which counteracts this driving force. This resting potential is defined as the potential at which the current across the membrane equals zero. The potential outside the ALC is considered to be ground and its value equals zero (i.e. VR=0). It is currently assumed that the resting potential of ions crossing through membrane leakage channels is zero (i.e. VRP=0). The resulting current due to membrane leakage, ileak, is given by ileak = (aki - VR - VRP)A = akiA
(2)
Stimulus channels are also modeled as continuously opened channels. The driving force of the current resulting from a given stimulus, I+, is the potential difference between the AR potential, aki, the potential outside the
4.2 AR Processing Figure 3 depicts the equivalent circuit modeling the artificial photoreceptor (AR) processing. This circuit reveals that membrane leakage, stimulus, the potential of its neighboring ARs connected by gap junctions, and the potential of its surrounding AECS affect AR potential. Applying current flow analysis to the AR node of this
Figure 3. Electrical circuit modeling photoreceptor axon (AR) processing.
artificial
ALC, VR, and the resting potential, B. As with leakage channels, the potential outside the ALC equals zero (i.e. VR=0). However the resting potential, B, is not equal to zero. In examining the effects of the stimulus-input value, it was concluded that an increased stimulation increases this current flow. This infers that the resistance of stimulus channels varies and is inversely proportional to the stimulus-input value. Therefore the resistance of stimulus channels is 1/I+. For a given stimulus input value, I+ki, the current as a result of stimulus is given by istimulus = (aki - VR - B) I+ki = (aki - B) I+ki
(3)
Gap junctions are electrical synapses that directly connect neurons. These channels conduct the flow of ionic current and mediate the electrical transmission between the neurons. The driving force of current through gap junctions is a diffusional force caused by the ionic concentration differences between the connected neurons. One can regard ions as moving down their concentration gradient, creating an electrical potential difference. The resulting potential difference is measured as the potential difference between the neurons connected by gap junctions and is proportional to the gap junction resistance, 1/D. Current through a single gap junction is igap = (a1 - a2)D (4) where a1 and a2 are the potentials of the connected neurons. Since each AR has a right and left neighboring AR connected by gap junctions, letting aL and aR represent the potentials of the left and right neighboring ARs respectively, the total current flowing through gap junctions is given by igap = (aL - aki)D + (aR - aki)D = (aL - 2aki + aR)D
(5)
Interactions between an AR and its surrounding AECS are assumed to be nonsynaptic and diffusive. As with gap junction channels, AR/AECS channels conduct the flow of ionic current and the driving force is a diffusional force proportional to a constant resistance. The diffusional force is measured as the potential difference between the AR and AECS potentials. Given the AR potential, aki, the potential of its AECS, zk, and a channel resistance of 1/E, the equation for AR/AECS current is iAR/AECS = (aki - zk)E
(6)
Substituting the above current equations into Equation 1 and dividing the AR capacitance, Ca through to solve for aki yields daki/dt = -akiA′ + (B′- akiC′ )I+ki + (aL - 2aki + aR)D′+ (zk - aki)E
(7)
The above nonlinear equation is the first order differential shunting equation used to calculate the changes to AR potential.
4.3 AECS Processing Figure 4 depicts the equivalent circuit modeling the artificial extracellular space (AECS) processing. This figure shows that AECS potential is affected by the potential of its contributing ARs as well as the potential of the AECS of its neighboring ALCs. Applying current flow analysis to the AECS node of this circuit gives Czdzk/dt = -iAECS/AECS + iAECS/AR
(8)
where Cz is the AECS capacitance, iAECS/AR is the current through all AECS/AR channels, and iAECS/AECS is the current through all AECS/AECS channels. As previously stated, interactions between an AR and its surrounding AECS are assumed to be nonsynaptic and diffusive. All channels in this circuit are presumed to be continuously open and the driving force of the current through these channels is a diffusional force. As before, the diffusional force is measured as the potential difference between its respective potentials. Unlike the AR potential that is affected by a single AECS potential, AECS potential is affected by its six contributing ARs. Given the potential for each AR, aki, the AECS potential, zk, and a channel resistance of 1/E, the equation for the total AR/AECS current is iAECS/AR = Σi(aki – zk)E
i=1,6
(9)
AECS potential is also affected by the AECS potential of its neighboring ALCs. Given the AECS potential, zk, the AECS potential of its N adjacent ALCs, zj, and a resistance of 1/L, the equation for the total current through AECS/AECS channels is iAECS/AECS = Σj(zk – zj)L
j=1,N
(10)
Substituting the above current equations into Equation 8 and dividing the AECS capacitance, Cz, through to solve for dzk/dt gives dzk/dt = -Σi(aki – zk)E - Σj(zk – zj)L (11) This equation is the first order diffusion equation used to calculate the changes to AECS potential.
Activation Differences
AL1 and AL2. Figure 5 depicts this difference for AR1 across the center row of ALCs used during narrow beam stimulus testing. Since all ARs in the center ALC were uniformly excited, similar results were obtained for the remaining ARs, AR2-AR7. Similar results were also found in examining the activation differences in the center column of the array. 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 -0.01 -0.02
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ALC Number
Figure 4. Electrical circuit modeling artificial extracellular space (AECS) processing.
5 Results Testing utilized a 9x9 array of ALCs. Each ALC contained seven ARs, as described in Section 3, with the six peripheral ARs, AR1-AR6, contributing to activation processing. To investigate the on-center off-surround response, the ANN was tested using a narrow beam illumination stimulus. For this, all ARs within the central ALC were uniformly excited while the ARs in the remaining ALCs received no stimulation. Initially all ARs within each ALC were illuminated with a background stimulus value of 0.5. Upon allowing the ANN to converge, the amount of stimulation for all ARs within the central ALC was increased to 1.0 while the ARs in the remaining ALC were illuminated with the same background stimulation of 0.5. Effectively, the stimulus values used to illuminate the ARs in the center ALCs received a step function type illumination. The initial value for all AR and AECS potentials was 0.5. Changes to AR potential, da/dt, were calculated using the nonlinear first order differential shunting equation, Equation 7. Shunting equations parameters were selected to allow a minimum activation value of 0.0 and a maximum activation value of 1.0 and the time constant was set to τ = 2.0. Changes to AECS potential, dz/dt, were calculated using the first order diffusion equation, Equation 11. Both equations were approximated using a 2nd Order Runga Kutta method. The difference between the converging values for the AR and AECS potentials is used as input into the ALMCs,
Figure 5. Narrow beam stimulus testing results. Shown is resulting potential difference between AR1 and its respective AECS for the center row of a 9x9 array of ALCs. Similar results were obtained the remaining ARs, AR2-AR6. To investigate the frequency response, the ANN was tested using a laser beam illumination stimulus. For this, all ARs within the central ALC were excited with a sinusoidal input while the ARs in the remaining ALCs received no stimulation. The initial value for all AR and AECS potentials was 0.5 and shunting equations parameters were selected to allow a minimum activation value of 0.0 and a maximum activation value of 1.0. An animation which depicts the array of ALCS is provided. This animation shows the excitation input into each ARs (upper left array), the result AR activation value as calculated using Equation 7 (upper right array), the activity diffusing into the AECS of each ALC (lower left cartridge) and the potential difference between the ARs and their respective AECS (lower right array).
6 Discussion Upon examination of the difference between the AR and AECS potentials for the given narrow beam, constant illumination stimulus, an on-center off surround response is evident. The asymmetric nature of the response is attributed the hexagonal arrangement of the ALC array. Although only the results for a single AR are depicted AR1, the same response was found in examining the potential differences between the remaining ARs and their respective AECS. It was concluded that due to nature of
stimulation used during testing, these responses were correct. It was surprising to note the response of the nonsynaptic ANN was similar to synaptic modeling in that excited ALCs laterally inhibited their surrounding ALCs. Further examination of testing results revealed that the AECS potentials in the surrounding ALCs were also inhibited. This indicates that the extracellular space surrounding a network of artificial neurons is a viable source of lateral communication within an artificial neural. Results also imply that in artificial systems, the functional role for nonsynaptic communication is similar to synaptic communication. If this were true of the early visual processing within the fly’s optic lobe, this provides an interesting homology between the visual processing of different species and exemplifies that computational principles can be implemented two difference ways in nature.
7 Conclusions A biologically inspired ANN modeling nonsynaptic communication between neurons has been presented. In this model, an AR’s potential is indirectly affected by the potential of its neighboring ARs as well as the potential of its surrounding AECS. The AECS potential diffuses into adjacent ALCs and the ARs encased in each ALC. Test results demonstrate that nonsynaptic communication between neurons can achieve results similar to synaptic connections. This implies that the ECS surrounding neurons should be considered as a source of lateral communication amongst a network of neurons.
8 Acknowledgements I would like to that Albuquerque High Performance Computer Center and the Department of Electrical Engineering at the University of New Mexico for their support in the development of this model. I would also like to thank Jo-Anne Green for her time and artistic expertise in developing figures for this paper.
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