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1 A Populaion Firing Rae Model of Reverberaory Aciviy in Neuronal Neworks Zofia Koscielniak Carnegie Mellon Universiy Menor: Dr. G. Bard Ermenrou Universiy of Pisburgh Inroducion: The field of mahemaics has made remendous impac on he sudy of neuroscience. The elecrochemical aciviy of neurons is precisely imed and execued and hus ideal for mahemaical modeling. Mahemaicians are able o model neural aciviy ha can be experimenally confirmed wih sarling accuracy. Mahemaical models also give insigh ino how cerain facors can affec neural aciviy. As compuer echnology improves and compuaion ime decreases mahemaicians are able o consruc more complicaed and realisic models. Today various approaches o modeling he aciviy of he nervous sysem exis. They include invesigaing elecrical and chemical aciviy inside an individual neuron as well as a nework of cells. All hese models provide valuable insigh ino how he nervous sysem funcions. The srucure of a neuron is specifically designed o propagae he spread of chemical signals. A he resing sae he cyoplasm inside he neuron has a slighly negaive charge. When he cell receives a signal of sufficien srengh from anoher cell hey i depolarized enough o allow a signal o ravel down he axon causing he neuron o fire. The elecrical poenial difference ravels down he lengh of he axon and causes he synapic vesicles o release a neuroransmier signal and poenially exciing he dendries of an adjacen neuron (3. A saed before he poenial inside a neuron has o be sufficienly increased before a signal can propagae down he axon. This iniial depolarizaion is caused by he neuroransmier signal form anoher neuron. These signals come in shor burss and he
2 depolarizaion is revered back o resing sae quickly. Anoher kind of signal called asynchronous synapic ransmission has been shown o elevae he poenial of he cell slighly and ake a much longer ime o reurn o resing sae hen he regular neuroransmier signal (5. This signal raises he poenial of he cell more han he regular signal would by iself and can cause he cell o reach he depolarizaion hreshold. The cell hen fires an acion poenial ha i would no have wihou asynchronous synapic ransmission. There in addiion o sources of exciaion here are also negaive feedback mechanisms which preven run-away aciviy in neworks. In my research I plan on developing a neural firing rae populaion model ha explores wo such sources spike frequency aion synapic depression and wheher hese in combinaion wih asynchronous release are sufficien o explain synchronous populaion burss in cell culure neworks. Mehods: Populaion Firing Rae Model A populaion firing rae model is based on deermining a funcion for he firing rae of a group of neurons given heir inpus. This funcion mean o characerize he probabiliy of acion poenials (4. The basic model begins wih a designaed se of imes 1 o n (represening he spike imes of a neuron and an empirically-deermined α- funcion ha represens he responses of pos-synapic cells o pre-synapic spikes. The oal response of he pos-synapic cell a a ime becomes α + ( 1 n α ( n j 1 + α (. Le equal he firing rae of he sysem such ha a any i. j The oal response of he sysem becomes he inegral
3 n Φ( α( ( pr( s ds where pr ( s j is he probabiliy ha s j. In a j 1 j populaion model however ha probabiliy equals he firing rae because i represens he poenial of a ypical neuron receiving a re-synapic spike so his inegral can be wrien as Φ( α ( ds. In any paricular synapse he firing rae of he possynapic cell is deermined by he firing paerns of he pre-synapic cell. Thus μ ( F( Φ ( F( α( μ pos pos pre ( d where he firing rae of a possynapic cell is a non-linear funcion of he response of ha cell based on he inpus i encouners. One of he fundamenal assumpions of a populaion model is ha he firing raes of all neurons are he same hus μ μ and he firing rae can be wrien as pre μ ( F α( ds. All populaion models use his basic problem srucure bu vary in he selecion of he α funcion. pos Mehods: A Populaion Firing Rae Model of Spike Frequency Adapaion My firs model of reverberaory aciviy will be based on spike frequency aion. Spike frequency aion is he gradual reducion of firing frequency of a neuron wih coninuous inpus (1. Insead of one response funcion Φ I will now have hree represening responses o regular signals asynchronous synapic ransmission and spike frequency aion. The firing rae now becomes μ F( AΦ (
4 where Φ α ( ds Φ α ( ds and Φ α ( ds F is a nonlinear funcion of inpus o he cell and AB and C are he maximum magniudes of corresponding responses. Since his model will represen a gradual decrease in firing frequency i makes sense for he α funcions o be decaying exponenially. Thus α e α e and α e where and are ime consans deermined from empirical daa. The firs fundamenal heorem of calculus can be used o deermine he firs derivaives of he response funcions. The derivaives become: + μ ( + μ ( and + μ (. As previously saed μ F( AΦ so hese funcions become ( and. Thus his sysem can be described wih hree firs-order differenial equaions. My research will include a full analysis of his sysem.
5 Mehods: A Populaion Firing Rae Model of Synapic Depression The synapic depression model is characerized by many componens of he spike frequency aion model. Shor-erm synapic depression refers o he observaion ha as a cell fires a some rae i does no have enough ime aferwards o fully recover o he neuroransmier level i had before (2. Leing σ represen he amoun of exhausible resources in he cell a any ime and f be he fracion of hose resources afer every fire ha is los change in σ( becomes dσ 1 σ f σ where dep is a ime dep consan deermined empirically. Assuming ha synapic depression only affecs he reacion and no asynchronous synapic ransmission becomes μ ( F( AσΦ reducing he sysem o hree differenial equaions: and dσ 1 σ ff( AΦ σ. dep Mehods: Analyzing Sysems of Differenial Equaions The firs sep in analyzing eiher of hese sysems is solving hem. From here i is necessary o find parameers ha cause he sysem o exhibi he oscillaing behavior seen experimenally. From here I plan on performing a bifurcaion analysis on ime consans ( maximum reacion consans (A B C and he depression consan (f. A bifurcaion analysis involves examining he consequences of varying parameers on he
6 sabiliy and seady saes of a sysem (3. Finally I plan o couple hese sysems ogeher and sudy how spike frequency aion and synapic depression affec each oher in he populaion model. A mahemaical model of reverberaory aciviy in neural neworks will provide powerful insighs ino he funcion of he sysem. References 1. Benda Jan Longin Andre and Len Maler. "Spike-Frequency Adapaion Separaes Transien Communicaion Signals From Background Oscillaions." The Journal of Neuroscience 25 (25: De La Rocha Jaime and Nesor Parga. "Shor-Term Synapic Depression Causes a Non-Monoonic Response o Correlaed Simuli." The Journal of Neuroscience 25 (25: Edelsein-Keshe Leah. Mahemaical Models in Biology. 1s ed. New York: The Random House/Birkhauser Mahemaics Series Ermenrou G. Bard. "Neural Neworks as Spaio-Terminal Paern-Forming Sysems." Repors on Progress in Physics 61 (1998: Lau Pak-Ming and Guo-Qiang Bi. "Synapic Mechanisms of Persisen Reverberaory Aciviy in Neuronal Neworks." Proceedings of he Naional Academy of Sciences of he Unied Saes of America 12 (25:
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