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3 Submitted in total fulfilment of the requirements for the degree of Doctor of Philosophy in Applied Mathematics THEORETICAL MODELING AND ANALYSIS OF NEURONAL DENDRITIC INTEGRATION SONGTING LI Supervised by Prof. DAVID CAI Prof. DOUGLAS ZHOU DEPARTMENT OF MATHEMATICS SHANGHAI JIAO TONG UNIVERSITY SHANGHAI, P.R.CHINA November 214

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5 献给我的家人

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11 ,.,,..,..,.,,.,.,.,, HH IF.,,.,., -.,.,.,,.,., - - -,.,,. i

12 ,.,.,.,IF., IF, DIF., DIF., DIF,.,.,, IF HH, DIF DHH., DHH, HH,..,,.,,.,,.,..,..,,.,,,,,, ii

13 ABSTRACT THEORETICAL MODELING AND ANALYSIS OF NEURONAL DENDRITIC INTEGRATION ABSTRACT A neuron, as a fundamental unit of brain computation, exhibits great computational power in processing input signals from neighboring neurons. It receives thousands of spatially distributed synaptic inputs from its dendrites and then integrates them at the soma, leading to the neuronal information processing. This procedure is called dendritic integration. Dendritic integration rules are under active investigation in order to elucidate information coding in the brain. In the thesis, we present our work on theoretical modeling and analysis of dendritic integration in the following six chapters. To be specific, we introduce the background knowledge in Chapter 1 and Chapter 2, and introduce our own research work from Chapter 3 to Chapter 6. In Chapter 1, we introduce basic neurophysiology for mathematicians who are not familiar with neuroscience. We also review the current progress in the experimental and theoretical investigation on dendritic integration. We finally point out the scientific contribution and novelty of our work. In Chapter 2, we introduce two types of neuron models to characterize neuronal electrophysiological properties described in Chapter 1. A neuron can be modeled as an idealized point with electrical circuit structure, such as the HH model and the IF model. On the other hand, a neuron can also be modeled as a spatially extended tree with conductive cable structure, such as the two-compartment and multi-compartment cable models. All these models can describe neuronal behavior effectively in different aspects, therefore, they will be used in our following theoretical study of dendritic integration. In Chapter 3, we reveal theoretically the underlying mechanism of a dendritic in- iii

14 tegration rule for a pair of excitatory (E) and inhibitory (I) synaptic inputs discovered in a recent experiment. Starting with the two-compartment neuron model, we construct its Green s function and carry out an asymptotic analysis to obtain its solutions. Using these asymptotic solutions, in the presence of E-I inputs, we can fully explain all the experimental observations. We then extend our analysis with multi-compartment neuron model to characterize the E-I dendritic integration on dendritic branches. The novel characterization is confirmed by a numerical simulation of a biologically realistic neuron as well as published experimental results. In Chapter 4, we theoretically generalize the dendritic integration rule in Chapter 3 to describe the spatiotemporal dendritic integration for all types of inputs, including a pair of E-I, E-E, I-I inputs and multiple inputs with mixed types. In addition, the general dendritic integration rule is valid at any time during the dendritic integration process for inputs with arbitrary arrival time difference. The general rule is derived analytically from the two-compartment neuron model. However, we also verify it in a simulation of the realistic neuron and in experiments. The general rule finally leads us to a novel graph representation of the dendritic integration process, which is demonstrated to be functionally sparse. In Chapter 5, we address the theoretical issue of how much the dendritic integration rule discovered in the experiment can be accounted for using the somatic membrane potential dynamics described by the point neuron model. We demonstrate both analytically and numerically that the IF model can explain partial of the experimental results. Inspired by a two-port analysis, we then modify the IF model to the DIF model to characterize all the experimental observations. Meanwhile, the DIF model provides experimental testable predictions. In Chapter 6, we systematically investigate the performance of the point neuron models in characterizing the spatiotemporal dendritic integration effect. We demonstrate numerically that, compared with the standard IF model and HH model, our DIF model and DHH model can accurately capture the membrane potential produced by the two-compartment neuron model with a passive or an active soma, respectively. In particular, our DHH model can accurately predict the spike time of the two-compartment neuron model, whereas the prediction error made by the HH model is significantly iv

15 ABSTRACT large. In addition, the HH model occasionally predicts a fake spike. The scientific contribution and the novelty of our work can be summarized as follows. First, the nonlinearity in the cable equation with time-dependent synaptic inputs makes its analytical solution difficult to obtain. Here we analytically solve the cable equation via the asymptotic analysis, and apply the asymptotic solutions to reveal the underlying mechanism of the dendritic integration rule discovered experimentally. In addition, the previous research work on dendritic integration are mainly qualitative and specific. Here we propose a general dendritic integration rule to quantitatively describe dendritic integration for all types of synaptic inputs. The general rule is further confirmed in the realistic simulations and real experiments. Moreover, point neuron models are considered only to describe the somatic membrane potential in previous works. Here we incorporate the dendritic integration effect into point neuron models successfully. Contrast to the cable model, our effective point neuron model can be potentially used in a large scale simulation of a network of neurons with dendrites to reduce the computational cost. KEY WORDS: dendritic integration, synaptic integration, neuronal computation, cable theory, Green s function, asymptotic analysis, point neuron model v

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17 ABSTRACT i iii vii xiii xv HH vii

18 2.1.2 IF viii

19 IF DIF DIF DHH ix

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21 IF κ EI xi

22 IF IF IF β κ M κ DIF DIF DIF DIF DIF DHH DHH DHH xii

23 6 8 DHH DHH xiii

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25 v v r v th g E f E ε E g I f I ε I g L ε L g Na E Na g K E K G E G I c S d l xv

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27 ,. [1, 2], (synapse),,..,,.. 1.1,,.,, 1 1.,., (dendrites), (soma) (axon), ,, 1 2. (presynaptic), (postsynaptic).,,, ( )., ,.. 1

28 神经元树突整合的理论模型与分析 上海交通大学博士学位论文 A B C D E F 图 1 1 神经元的多样性. (A) 椎体神经元, 记录自猕猴前额叶皮层 [3], 长约 696µm. (B) 星型 神经元, 记录自小鼠内嗅皮层 [4], 长约 39 µm. (C) 蒲氏神经元, 记录自大鼠小脑 [5], 长约 118µm. (D) 篮状神经元, 记录自大鼠体感皮层 [6], 长约 267µm. (E) 颗粒神经元, 记录自大鼠 海马区 [7], 长约 288µm. (F) 运动神经元, 记录自猫的脊髓 [8], 长约 1.5mm. 以上数据均下载 自 NeuroMorpho.Org [9]. Fig 1 1 The diversity of neuronal morphology. (A) A pyramidal cell in macaque prefrontal cortex with a length of 696µm [4] [3]. (B) A stellate cell in rat entorhinal cortex with a length of 39 µm. (C) A Purkinje cell in mouse cerebellum with a length of 118µm [5]. (D) A basket cell in rat somatosensory cortex with a length of 267µm [6]. (E) A granule cell in rat hippocampus with a length of 288µm [7]. (F) A motor neuron in cat spinal cord with a length of 1.5mm [8]. All data is downloaded from NeuroMorpho.Org [9]. 2

29 1 2..,,,,.,,.,. [1]. Fig 1 2 The typical structure of a neuron. The postsynaptic neuron receives synaptic inputs from a presynaptic neuron. When an action potential is initiated and propagated to the axon terminal of the presynaptic neuron, neurotransmitters will be released. These neurotransmitters will bind to some specific channel receptors and invoke the open of the ion channels in the postsynaptic neuron. Consequently, the ionic currents will flow inward or outward the membrane thus change the local membrane potential of the postsynaptic neuron. The ionic currents inside the postsynaptic neuron will then flow towards the soma along the dendrites. In the end, the soma integrates all synaptic inputs and generates an action potential at the axon hillock once its membrane potential crosses a certain threshold. The action potential will then prorogate along the axon towards the downstream neurons. Figure is modified from Ref. [1]. 3

30 , [11 13] ,. (threshold), (axon hillock) 1ms 2ms, 1mV, (action potential) (myelin),,, 1mV.,. (receptor),,. (chemical synapse)., ( ). 1.2,., (ion channel),. (gated) (nongated)., (leak).,,..,,.,,.,,,,. (reversal potential).,. 4

31 . x [C](x), ϕ(x), (Fick s law), J diff J diff = D [C] x, D,. J diff..,,,. J drift J drift = µz[c] ϕ x. µ (mobility), z (valence)., J total = D [C] x µz[c] ϕ x., D µ, D = k BT q µ, k B, T, q. J total = k BT q µ [C] x µz[c] ϕ x. (1 1) (1 1) Nernst-Planck., ( ). ϕ x = k BT [C] zq[c] x. Nernst (1 2) E ϕ in ϕ out = k BT zq ln [C] in [C] out. (1 2) 5

32 E. K + [14],, 5mM, 14mM. 37 o C, k B T /q = 26.73mV, E K = 62 ln ( 14) = 89.7mV. 5,, -7mV. (resting potential).,.,,,,. (hyperpolarization)., E Na = +55mV.,,,,. (depolarization).,, , -7mV.,., Nernst-Planck (1 1).,,,.,, Nernst-Planck.,, Goldman, Hodgkin Katz [15, 16]. Nernst-Planck 6

33 v r, ϕ(x) x, l, dϕ/dx = v r /l., µ., η,, [C], η[c]., Nernst-Planck J = µ k BT q η [C] x µ zη[c] v r l, < x < l [C]() = [C] in, [C](l) = [C] out. [17] ( e λ [C] out [C] ) in J = P λ, e λ 1 P = µ ηk B T lq (permeability).. λ λ = zqv r k B T. Na +,K + Cl,,, J Na + J K + J Cl =. v r = k BT q ( PNa [Na] out + P K [K] out + P Cl [Cl] ) out ln P Na [Na] in + P K [K] in + P Cl [Cl] in (1 3) 7

34 (1 3) Goldman-Hodgkin-Katz (GHK). [14], P K : P Na : P Cl = 1 :.3 :.1, [K] in = 4mM, [Na] in = 5mM, [Cl] in = 4mM, [K] out = 1mM, [Na] out = 46mM, [Cl] out = 54mM. (1 3) 74mV,. GHK (1 3), Nernst (1 2). 1.4.,,. ( -55mV), [18, 19]...,. 1 3,,,.,. 1.2,,,.,,,.,.,,. 1ms 2ms, 1mV.,,,. (refractory period).,,,,., 8

35 E Na V Na + channels close Na + channels open K + channels open Threshold V R E K Refractory period time 1 3.,,.,,,. [14]. Fig 1 3 The action potential. During the upstroke, Na + channels are open and the membrane potential approaches the Na + reversal potential. During the downstroke, Na + channels are closed, K + channels are open, and the membrane potential approaches the K + reversal potential. Figure is modified from Ref. [14]. 9

36 ,., ,.,,.,. (EPSP).,,. (IPSP).. (glutamate), AMPA NMDA. AMPA. NMDA,. EPSP, mv., AMPA, NMDA.,,,., AMPA. γ- (GABA), GABA A GABA B. GABA A,, -8mV,, IPSP. GABA A, IPSP,., GABA B,,,.,, -1mV., GABA A, GABA B. 1

37 1.6 [2].,,, [21] [22]., [23, 24], [12, 25].,,,,.,.,,..,, [12, 26].,.,,. (passive).,,,.,.,.,,,., [27]., EPSP EPSP [27]., EPSP EPSP., 11

38 Excitatory Input EPSP EPSP EPSP ( ). +,. Fig 1 4 Passive dendritic filtering effect. Activation of a excitatory input on an apical dendrite produces a local EPSP that is larger and faster than the EPSP recorded at more proximal locations and at the soma (as indicated by recording electrode symbols). The ion concentration is indicated by + and the current flow is indicated by arrows , [28]., 1-2µm, 2µm.,,.,,,.. [26]..,, (Summed Somatic Potential, SSP). 12

39 ,, EPSP,., EPSP, EPSP..,,.,, (driving force), EPSP.,, EPPS.,,, [29], [3, 31] [32]. [33]., [34]., [35, 36]. H, [37]... [38, 39] [19].,,.,.,,.,,,,.,,.,., 13

40 ,,,,,. 4:1 [4].., [41 44]., [45] γ [46].., [47], [48] [49], [5 52]. [53].,, (hyperpolarization) (shunting inhibition).,.,. [54, 55].,,,. [55, 56]. [57],, IPSP,,, , [17, 58].,.,,. 14

41 Excitatory input Inhibitory input EPSP IPSP SSP 1 5. ( ) ( )., IPSP ( ).,, ( SSP) EPSP ( ) IPSP ( ).. Fig 1 5 The interaction between a pair of excitatory and inhibitory inputs. A single excitatory synapse (circle) on an apical dendrite is shunted by an inhibitory synapse (triangle) on the path between the synapse and the soma. Note that the individual IPSP amplitude is nearly zero (middle) when the inhibitory input is given alone. However, when both inputs are given, the summed somatic potential (SSP on the right) is much smaller than the sum of the individual EPSP (left) and IPSP (middle). Three recording sites are indicated by recording electrode symbols.,.,, [58, 59].,.,Wilfrid Rall [6], (cable theory) [61, 62].,,.,..,, [63] 15

42 ,.,., [58].,., [64, 65],,.,,., [61].., :EPSP, [66, 67]. [68, 69].,, [7] , :,,,.,,.,,.,, -,,,..., ( 2.1)., 16

43 . PDE, ODE,. 17

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45 .,., ,.,, ,,.,.,. 2 1,.,,., I cap = dq dt, 19

46 A C v B g V reversal I inj c 2 1. (A). 3-5nm.. [71]. (B).,.,,.,. (C). Fig 2 1 Equivalent circuit model. (A) Schematic representation of a small patch of typical membrane. The 3-5nm thin bilayer of lipids isolates the extracellular side from the intracellular side. From an electrical point of view, the resultant separation of charge across the membrane acts as a capacitance. Ion channels inserted into the membrane provide a conduit through the membrane. Figure is modified from Ref. [71]. (B) The associated electrical circuit for this patch consisting of a capacitance and a conductance in series with a battery. The conductance mimics the behavior of ion channels inserted throughout the membrane and the battery accounts for the ion channel s reversal potential. Note that here we only consider one type of ion channel, however, multiple types of channels can be connected in parallel. (C) The electrical circuit representation for the whole membrane. 2

47 q. c, v, q = cv. I cap = c dv dt. (2 1),., c = 1µF/cm 2[72]., [73] I leak = g L (v ε L ), (2 2) g L ( ). ε L.,., I inj,, I cap + I leak = I inj. c dv dt = g L(v ε L ) + I inj. (2 3), I Na I Na = g Na (v E Na ), g Na, E Na. I K I K = g K (v E K ), g K, E K. (2 3) c dv dt = g L(v ε L ) g Na (v E Na ) g K (v E K ) + I inj. (2 4) g L,, g Na,g K v, g L v., (2 4),. g Na g K

48 2.1.1 HH HH Hodgkin Huxley 1952 [74], , HH., c dv dt = g L(v ε L ) g Na (v E Na ) g K (v E K ) + I inj. g Na g K. g Na g K,. (voltage clamp),., I inj [75]., v, g Na g K, I cap = cdv/dt.,. g L.,,,. I inj,, g L ε L. I inj g L (v ε L ). (2 5) mv. 2 2,, I inj,..,,.,. Hodgkin Huxley., 22

49 I K I inj I Na Time (ms) -65 mv mv 2 2. mv.,.. Fig 2 2 Numerically computed voltage-clamp experiment. The membrane potential is stepped from the resting potential to mv. This result indicates an inward current followed by an outward current. The K + and Na + currents are also shown here. 23

50 ., I inj I K.,, g K g K (v) = I K v E K.,, I Na = I inj I K. g Na g Na (v) = I Na v E Na. 2 3 g Na g K 4. g Na g K.,, g Na., g K.., ( ) m ( ) h.., m h., m,., h,,.,, Hodgkin Huxley g K = ḡ K n 4, g Na = ḡ Na m 3 h, (2 6) ḡ K ḡ Na, n,m h,, 1. n 4 :,. m 3 h :.,. 24

51 4 V C (mv) 4 g 2 2 Na (ms / cm ) (ms / cm ) g K Time (ms) Fig 2 3 The voltage dependence of the sodium and potassium conductances in simulation. The membrane potential is stepped to different values and the resulting K + and Na + conductances are obtained numerically (m,n h),,,., C α(v) O, β(v) C O, α(v), β(v),. r (r = m, n, h), 1 r,, dr dt = α r(v)(1 r) β r (v)r. (2 7) 25

52 (2 7) dr dt = 1 τ r (v) (r r), (2 8) (2 8) r = τ r = α r (v) α r (v) + β r (v), 1 α r (v) + β r (v). r(t) = r (v) + (r() r (v))e t/τr(v), (2 9) r(). (2 9), v, r τ r (v) r (v)., Hodgkin Huxley, α n (v) =.1(v + 55) 1 exp( (v + 55)/1), β n (v) =.125exp( (v + 65)/8), α m (v) =.1(v + 4) 1 exp( (v + 4)/1), β m (v) = 4exp( (v + 65)/18), α h (v) =.7exp( (v + 65)/2), β h (v) = exp( (v + 35)/1). HH ḡ Na = 12mS/cm 2, ḡ K = 36mS/cm 2, E Na = +5mV, E K = 77mV, g L =.3mS/cm 2, ε L = 54.4mV. 26

53 steady state 1 h.8.6 m.4.2 n V (mv) (ms) 8 h 6 n 4 2 m V (mv) Fig 2 4 Gating variable functions. Left is the steady-state opening of the gates and right is their corresponding time constants. 2 4, n m,, mv 1.,, n m., h,,., τ m τ n τ h,,,, IF HH,.,,.,,. HH - (Integrate-and-Fire, IF) [73] c dv dt = g L(v ε L ) + I inj. (2 1) IF (2 1),. v v th,, v ε L,

54 -55 mv -7 mv na Time (ms) IF. IF 2nA., -7mV, -55mV. ( ) IF, ( ). Fig 2 5 Response of the IF model to a 2nA step current. The resting potential is set to be -7mV and the threshold is set to be -55mV. The action potential (marked by grey) is removed from the IF model and the voltage at the threshold (marked by red) is reset to the resting potential immediately. IF 197 [73]., IF [76, 77]. IF c = 1µF/cm 2, g L =.5mS/cm ,,. 1.6,,.,., Wilfrid Rall [6],.,,,. 28

55 2.2.1 (compartment), 2 6. [17, 72],. [x, x + x], d, πd x.,, πd x(i cap + I leak I inj ) = I long (x) I long (x + x). I long. I cap I leak (2 1) (2 2), cπd x v t = g Lπd x(v ε L ) + πd xi inj + I long (x) I long (x + x). (2 11) I long. [x, x + x], R a R a = r a 4 x πd 2. r a., [x, x + x] x, I long = πd2 v(x) v(x + x). 4r a x I long (x) = πd2 4r a v x. (2 12) (2 12) (2 11), cπd x v t = g Lπd x(v ε L ) + πd xi inj πd2 v 4r a x + πd2 v x 4r a x. (2 13) x+ x (2 13) x, c v t = g L(v ε L ) + I inj + d 4r a 2 v x 2. (2 14) (2 14),.. 29

56 I cap (x,t) I leak (x,t) I inj (x,t) d I long (x,t) I long (x+ x,t) x x x+ x ( ),. Fig 2 6 Passive cable model. Upper, a schematic plot of a neuron. Lower, the cable model with different current sources representing a small segment of the neuron s dendrites (marked by the black box). 3

57 A B C 2 7.(A). (B). (C). Fig 2 7 Spatial neuron model. (A) A schematic plot of a pyramidal neuron. (B) Two-compartment model. (C) Multi-compartment model (two-compartment). 2 7.B, l d, S.,.,,.,, (2 14) x = l v x =. (2 15) x=l,,, cs vs t = g LSv s + I dend, v s, I dend, (2 12) x =., 31

58 v s (t) = v(, t), x = v(, t) c = g L v(, t) + πd2 v t 4Sr a x. (2 16) x= 2.2.3,,, (multi-compartment). 2 7.C,, ,2,...,n + 1 l 1,l 2,...,l n+1, d 1,d 2,...,d n+1. 1 n n + 1., πd2 n+1 v n+1 4r a = xn+1 =l n+1 x n+1 n j=1 πd 2 j 4r a v j x j xj =. v n+1 (l n+1, t) = v 1 (, t) = v 2 (, t) =... = v n (, t). (2 17), (2 15),, (2 16). 32

59 ,. 1.6.,,., [59].,, [59]., [59], [59]. 3.1 [59]. 3 1.A., EPSP,.,, IPSP,.,, (summed somatic potential, SSP). [59], SSP EPSP IPSP (linear sum), 3 1.B. SSP EPSP+IPSP (shunting component, SC), V SC (t) V S (t) V E (t) V I (t), (3 1) V SC, V S, V E, V I SSP, EPSP, IPSP. EPSP t p, EPSP, IPSP SSP V E (t p ),V I (t p ) V S (t p ). 33

60 A B EPSP Linear sum SSP IPSP SC 5 ms 2 mv C D 2 (mv) SC 4 2 E: µm I : 53-7 µm 2 4 IPSP (mv) (mv) SC 1 E: µm I: 8-1 µm 5 1 EPSP (mv) [59]. (A). ( 232µm 129µm). 1µm. (B), EPSP, IPSP, SSP, SC, Linear sum. SC SSP Linear sum. (C) EPSP (9-1mV), SC IPSP.. R=.974. (D) IPSP ( mV), SC EPSP.. R=.965. [59]. Fig 3 1 Dendritic integration of a pair of excitatory and inhibitory inputs in experiments [59]. (A) Image of a rat CA1 pyramidal neuron. Arrows indicate excitatory and inhibitory input locations (at 232 and 129µm from the soma, respectively). (Scale bar: 1µm.) (B) Examples of EPSP, IPSP, SSP, SC and their linear sum response to a pair of concurrent excitatory and inhibitory inputs. SC is defined as the difference between the SSP and the linear sum. (C) SC vs. IPSP amplitude, measured for a fixed EPSP amplitude (9 1 mv). Data are from four cells. Line indicates linear fit (R=.974). (D) SC vs. EPSP amplitude, measured for a fixed IPSP amplitude ( mv). Data are from four cells. Line indicates linear fit (R=.965). Figures are modified from Ref. [59]. 34

61 [59],,,, t p, V SC (t p ) V I (t p ) ( 3 1.C), V SC (t p ) V I (t p ).,,, t p, V SC (t p ) V E (t p ) ( 3 1.D), V SC (t p ) V E (t p )., t p SC IPSP EPSP, SC EPSP IPSP, [59], V SC /V I V E, V SC /V E V I.,, 3 2.A. V SC (t p ) = κv E (t p )V I (t p ). κ, EPSP IPSP., V SC (3 1), t p, V S (t p ) = V E (t p ) + V I (t p ) + κv E (t p )V I (t p ). (3 2) [59] κ. 3 2.B,, κ,, κ,. 3.2 [59].,.., (i) κ,(ii) κ. κ. 35

62 ( or S C / I P SP ) S C / E P SP A IPSP (or EPSP ) (mv) - 1 ) mv ( κ B I (µm) E location (µm) (A) EPSP, SC EPSP IPSP ( ) SC IPSP EPSP ( ).. EPSP 1-1mV, IPSP.2-4mV µm. : R=.96, κ=.142, n=11; R=.92, κ=.145, n=1. (B) κ., κ,, κ.. [59]. Fig 3 2 Dendritic integration rule for a pair of excitatory and inhibitory inputs in experiments. (A) Ratio between measured SC and EPSP (SC/EPSP) plotted against IPSP (red circle) and SC/IPSP plotted against EPSP (blue square) at the time when EPSP reaches its peak value. Data are from the same cell in the slice recording. The amplitudes of the paired EPSP and IPSP were randomly set in the range of 1 1mV and.2 4mV, respectively. Excitatory and inhibitory input locations were fixed at 11µm and 45µm. Lines indicate linear fit (red: R=.96, slope κ=.142, n=11; blue: R=.92, slope κ=.145, n=1). (B) The shunting coefficient κ as a function of the excitatory input location. for a fixed location of the inhibitory input on the dendritic trunk, κ increases as the distance between the excitatory input and the soma increases when the excitatory input is located in between the soma and the inhibitory input, whereas κ remains almost constant when the excitatory input is located further away from the soma than the inhibitory input. Three different inhibitory input locations are marked by different colors. Figures are modified from Ref. [59]. 36

63 3.2.1,. I syn [63], I syn = g syn (v ε syn ). (3 3) g syn, v, ε syn. g syn,,. g syn [72] g syn = fg(t), f, g(t), g(t) = N(e t σ d e t σr )Θ(t), (3 4) σ r g(t), σ d g(t), Θ(t) Heaviside,, N g(t) 1, ( σr ) σr σ N = [ d σr σ d ( σr ) σd σ d σr ] 1. σ d,,,. ( B),, l, d., -., cπd x v t = g Lπd xv + I syn + I long (x) I long (x + x), (3 5) v ( v = mv), c, g L. I syn, I syn = πd q=e,i x+ x x 37 G q (v ε q )dx,

64 G E G I, ε E ε I. x = x E, x = x I, G E (x, t) = f E g E (t)δ(x x E ), G I (x, t) = f I g I (t)δ(x x I ). f E f I. g E g I (3 4), t g E (t) = N E (e σ Ed e t σ Er )Θ(t), g I (t) = N I (e t σ Id e t σ Ir )Θ(t) I long (2 12), (3 5) x, c v t = g Lv q=e,i f q g q (t)δ(x x q )(v ε q ) + d 4r a 2 v x 2. (3 6) r a , v x =, (3 7) x=l v(, t) c t = g L v(, t) + πd2 4Sr a v x. (3 8) x= S., v(x, ) =. (3 9) 3.2.3, (3 6) (3 7)-(3 9)., δ, G(x, y, t) c G t = g LG + d 2 G + δ(x y)δ(t), (3 1) 4r a x2 38

65 G(, y, t) c = g L G(, y, t)+ πd2 G(x, y, t) t 4Sr a x, x= G x =, and G(x, y, ) =. x=l, τ = t/c, ξ = x 4r a /d, η = y 4r a /d, λ = l 4r a /d, (3 1) H τ = g LH + 2 H + δ(ξ η)δ(τ), (3 11) ξ2 H(, η, τ) τ = g L H(, η, τ)+γ H(ξ, η, τ) ξ, ξ= H ξ =, and H(ξ, η, ) =, ξ=λ γ = πd2 r a d. 2S (3 11), LH(ξ, η, s) = A(η, s)e s+g L (ξ λ) + B(η, s)e s+g L (λ ξ) + e s+gl ξ η 2, (3 12) s + g L ( B(η, s) ), 1 s+gl [A(η, s) cosh( s + g L (λ ξ)) sinh( s + g L (η ξ))] for ξ η, LH(ξ, η, s) = 1 s+gl A(η, s) cosh( s + g L (λ ξ)) for ξ > η, (3 13) A(η, s) = (s + g L) sinh( s + g L η) + γ s + g L cosh( s + g L η) (s + g L ) cosh( s + g L λ) + γ s + g L sinh( s + g L λ). (3 14), (3 14) ζ(s). (3 13), (3 13), ζ(s) =. 39

66 ζ(s) =, LH(ξ, η, s). LH(ξ, η, s) LH(ξ, η, s) = n H n (ξ, η) s + k n, (3 15) H n (ξ, η) s, s = k n. (3 15), H(ξ, η, τ) = n H n (ξ, η)e k nτ. (3 16), (3 11), (3 16) k n H n (ξ, η). s = k n. w n = i k n + g L λ, ζ(s) = tan(w n ) = w n γλ, (3 17)., n 1, (3 17) w n (n 1/2)π < w n < (n + 1/2)π ; n =, w =. H n (ξ, η). s = k n C n,,., LH(ξ, η, s) ( 1 LHds = 2πi s C n LH (3 13)-(3 15) (3 18), ) 1. (3 18) s= kn H n (ξ, η) = γd n cos [w n (1 ξ/λ)] cos [w n (1 η/λ)], (3 19) D n = 2 [γλ + γλwn 1 sin(w n ) cos(w n ) + 2 cos 2 (w n )] 4

67 n. (3 1) G(x, y, t) G(x, y, t) = 4ra H(ξ, η, τ). (3 2) c 2 d I inj, (3 6). v(x, t) = G(x, y, t) I inj (y, t) (3 21),., (3 6) v,., (3 6),. (3 6), EPSP ( 5mV ), f E., IPSP ( -2mV ), f I. x E x I, v(x, t; x E, x I ) f E f I v = fe m fi n v mn (x, t; X ), (3 22) k=m+n=k X {x E, x I }. x E X m ; x I X n. (3 22) (3 6),,. O(1), c v t = g L v + d 4r a 2 v x 2. (3 23) (3 7)-(3 9), v =. (3 24),,. 41

68 O(f E ), c v 1 t = g L v 1 + d 4r a 2 v 1 x 2 + g E(t)δ(x x E )ε E, (3 25) (3 25) I syn = g E (t)δ(x x E )ε E,, (3 25) v 1 = G(x, x E, t) [ε E g E (t)]. (3 26).. (3 26) x E ε E g E (t), mv x E I syn = g E (t)( ε E )., v 1 v, v x E. O(fE 2 ), c v 2 t = g L v 2 + d 4r a 2 v 2 x 2 g E(t)δ(x x E )v 1. (3 27) (3 27) I syn = g E (t)δ(x x E )v 1, v 1 (3 26), (3 27) v 2 = G(x, x E, t) [ g E (t)v 1 (x E, t; x E )]. (3 28). (3 28) x E I syn = g E (t)v 1 (x E, t; x E ). 42

69 , v 2 v 1, v 1 x E.., O(f I ), O(fI 2 ), O(f E f I ), v 1 = G(x, x I, t) [ε I g I (t)]. (3 29) v 2 = G(x, x I, t) [ g I (t)v 1 (x I, t; x I )]. (3 3) c v 11 t = g L v 11 + d 4r a 2 v 11 x 2 g E(t)δ(x x E )v 1 g I (t)δ(x x I )v 1, (3 31) v 11 = G(x, x E, t) [ g E (t)v 1 (x E, t; x I )] + G(x, x I, t) [ g I (t)v 1 (x I, t; x E )]. (3 32) (3 6)., Crank-Nicolson,.1ms, 1µm. [58, 59], 3 1.,, 3 3., EPSP V E f E v 1 + fev 2 2, (3 33) IPSP SSP V I f I v 1 + f 2 I v 2, (3 34) V S = V E + V I + V SC, (3 35) V SC [59], V SC f E f I v 11. (3 36) 43

70 3 1. Table 3 1 Parameters for two-compartment neuron model. c 1. µf cm 2 g L.5 ms cm 2 ε L / mv ε E 7 mv ε I -1 mv S 9π µm 2 r a 1 Ω cm l 6 µm d 1 µm σ Er 5 ms σ Ed 7.8 ms σ Ir 6 ms σ Id 18 ms 3.2.5, (3 33)-(3 36)., SC V SC, (3 36)., κ = V SC V E V I v 11 (, t; x E, x I ) v 1 (, t; x E )v 1 (, t; x I ). (3 37) ( (3 33)-(3 36)), 3 3. (3 37) κ.,, f E f I EPSP IPSP. (3 37), κ (leading order) f E f I, κ EPSP IPSP. (3 2), κ (3 37). 44

71 A B C EPSP (mv) st 2nd n.s. 5 1 Time (ms) IPSP (mv) st 2nd n.s. 5 1 Time (ms) SSP (mv) st 2nd n.s Time (ms) 3 3. (A)EPSP. (B) IPSP. (C) SSP.,, (3 6) Fig 3 3 Asymptotic solutions of various orders for the two-compartment cable model (3 6) for (A) EPSP, (B) IPSP, and (C) SSP in comparison with numerical solutions of Eq. (3 6). The dashed blue line is the first order approximation. The red circle is the second order approximation. The black solid line is the numerical solution of the full Eq. (3 6). Parameters in our simulation can be found in Table 3 1. κ EPSP IPSP, 3 4.A. [59] κ, 3 2. ε E = 7mV ε I = 1mV, (3 26),(3 29) (3 32), (3 32) G(x, x E, t) [ g E (t)v 1 (x E, t; x I )]. (3 38) (3 32) v 11 v 11 G(x, x I, t) [ g I (t)v 1 (x I, t; x E )], (3 39) I syn = g I (t)v 1 (x I, t; x E ). x I v 1 (x I, t; x E ). (3 39), x I, κ κ G(, x I, t) [ g I (t)v 1 (x I, t; x E )]. (3 4) v 1 (, t; x E ) 45

72 A B SC/EPSP (or SC/IPSP) Experiment ) mv ( κ.2.1 Experiment IPSP (or EPSP) (mv) E location (µm) (A) (3 6) (3 2) SC EPSP (SC/EPSP) IPSP ( ) SC IPSP (SC/IPSP) EPSP ( ). EPSP IPSP (.2mV-3mV), SC/EPSP IPSP., IPSP EPSP (1mV-8mV), SC/IPSP EPSP. x E = 3µm, x I = 24µm.. (B) (3 6) κ : 5µm, 2µm 35µm, κ.. [59], x-y, Fig 3 4 Numerical experiments on dendritic integration rule for a pair of E-I inputs. (A) Simulation results of the two-compartment neuron model (3 6) in confirmation of the rule (3 2): Ratio of SC to EPSP (SC/EPSP) plotted against IPSP (red circle) and SC/IPSP plotted against EPSP (blue square). Fixing EPSP amplitude while varying IPSP amplitude from.2mv to 3mV, SC/EPSP increases linearly with IPSP. Similarly, fixing IPSP amplitude while varying EPSP amplitude from 1mV to 8mV, SC/IPSP increases linearly with EPSP. Lines indicate linear fit. Stimuli are given at x E = 3µm, x I = 24µm. Inset: experimental results (the inset is modified from Ref. [59]). (B) Spatial asymmetry of shunting coefficient κ in the model (3 6): κ as a function of distance between E location and the soma for three fixed I locations at 5µm, 2µm and 35µm, respectively (marked by colored lines). Inset: experimental results for the same set of the I locations (the inset is modified from Ref. [59]). The insets have the same axis labels as in the main figures. Parameters in our simulation can be found in Table

73 l 1, (3 17) w =, w n (n 1 2 )π n 1. k = g L, k n αw 2 n/l 2 n 1, α = d/4r a. (3 2) (3 26), v 1 (x I, t; x E ) v 1 β 1+γl e g Lt/c g E + 2βclg E αγ 1 w 2 n=1 n ( cos [w n 1 x I l )] ( cos [w n 1 x E l )]. (3 41) β = γε E 4ra c 2 d., 2 x E v 1 (x I, t; x E ) 2 v 1 x 2 E βcg E αγl n=1 = βcg E αγl = βcg E αγl [ ( xi + x E cos 2l [ ( ) xi + x E cos nπ 2l n=1 ( xi x E cos 2l [ ( ) xi + x E δ c 2l ) nπ + cos ) ( xi x E (2n 1)π + cos 2l ( xi + x E cos l δ c ( xi + x E l ( xi x E l ) ) nπ )] nπ δ c ( xi x E 2l ) )] (2n 1)π ( )] xi x E + δ c, l δ c 2 [78] (Dirac Comb). x E,I [, l], x E = x I δ., xe v 1 (x I, t; x E ) x E. (3 41), xe v 1 (x I, t; x E ) = (3 42) x E = l. x E [, x I ], xe v 1 (x I, t; x E ) x E [x I, l], xe v 1 (x I, t; x E ) =., 47

74 v 1 (x I, t; x E ), x E, x E = x I, x E > x I., v 1 (, t; x E ) x E [, l]. (3 4), l 1 κ. κ l. 3 4.B,., κ. x I, G(, x I, t) g I (t). x E [, x I ], x I v 1 (, t; x E ) v 1 (x I, t; x E ). x E x I, x E, v 1 (, t; x E )., x E x I, v 1 (x I, t; x E )., x E [, x I ], (3 4), κ x E. x E > x I, x E x I, v 1 (, t; x E ) v 1 (x I, t; x E ). (3 4), κ., (3 2) EPSP t p, [59].,...,., ( 2.2.3).,, ,, (3 2)., κ : 3 5.A, ( 3 5.A ), 48

75 A B.16 Branching Point C.16 I soma I I E I S I E, E I E κ (mv 1 ) E distance (µm) κ (mv 1 ) I 3 5. (A). (B) κ. ( ), ( ). ( ). (C) ( ), κ. κ. Fig 3 5 Spatial dependence of κ. (A) A schematic branched dendrite. (B) Spatial profiles of κ as a function of the E input location along the trunk of a realistic neuron (marked orange in the inset) for a fixed I input on a branch (red square). The dashed vertical line indicates the location of the branching point (green dot) along the trunk. (C) κ values (color-coded) for an I input (red dot) fixed at the apical trunk, with an E input scanned throughout the active dendrite. ( 3 5.A ).,,., κ x E ;, κ., κ. x E, x E x I,. κ (3 4), κ x E x E, (3 4), x I,κ v 1 (, t; x E ) v 1 (x I, t; x E ). x E ( 3 5.A ), I E, v 1 (x, t; x E ). E ( 3 5.A ), I E. E, I E,, I S, x I, I I. x E E 49

76 E I E, x I I S I I.,, v 1 (, t; x E ) x I v 1 (x I, t; x E ). (3 4), E E, κ ,. [59]., NEURON, Crank-Nicolson. CA1, Duke-Southampton Archive [79], 2.. : g Na g Kd A g p K A g d K A H g h AMPA,NMDA,GABA A,GABA B. [8 82]. AMPA,NMDA, GABA A dr dt = α[c](1 r) βr r, 1.α β,[c]. I syn = g syn (v ε syn ), (3 43) 5

77 v, ε syn, g syn. AMPA GABA A, ḡ ; NMDA, g syn = g syn = ḡr, (3 44) ḡr, (3 45) [Mg 2+ ]e.6v [Mg 2+ ]. GABA B,, g syn = ḡgn G n + K D, (3 46) G G-, n G-, K D. G dr dt = K 1[C](1 r) K 2 r, (3 47) dg dt = K 3r K 4 G, (3 48) K 1, K 2,K 3, K 4 G-. [66, 67, 8 82]. [59]. [83]. [34]. A, 35µm [35, 36]. H, [37].AMPA, [84 87]., g Na = 3mS/cm 2, g Na = 6mS/cm 2, g Kd = 5mS/cm 2, A x 1µm, ( g p K A (x) = ḡ KA 1 + x ), (3 49) 7 51

78 1µm < x 35µm, ( gk d A (x) = ḡ KA 1 + x ), (3 5) 7 x > 35µm g d K A (x) = 6.5 ḡ KA, (3 51) ḡ KA = 5mS/cm 2. H g h (x) = g s + g e g s 1 + exp[(l 2x)/(2δl)], (3 52) g s = 2µS/cm 2 (s ), g e = 1g s (e ), l = 6µm (l ), δl = 5µm. ( ) NMDA AMPA GABA B GABA A.6. R(x) = x. (3 53) l/2 : ( ) r m = r s + r e r s 1 + exp[(l 2x)/(2δl)], (3 54) r a = 8Ωcm, c = 1µF /cm 2, T = 34 o C, v r = 7mV. : E Na = +55mV, E K = 9mV, E h = 3mV, E AMP A = E NMDA = mv, E GABAA = 8mV, E GABAB = 9mV. 3 5.B, ( ),, κ, ( 3 5.B )., 3 5.C, ( 3 5.C ),κ. 3 5.B 3 5.C. 52

79 3.3.3 κ, [59] A,,., κ..,, κ. 3 6.B,,., κ κ..,κ, κ. 3 6.C,,,., κ..,, κ. 3 6.D,,,., κ κ..,, κ. 53

80 A κ (mv 1 ) EI.3.25 I E 2 E1 B κ (mv 1 EI ).3.25 I E E E1 E2.2 E1 E2 C κ (mv 1 ) EI.3.2 E 1 I E 2 D κ (mv 1 ) EI.3 E2 I E E1 E2 E1 E [59].,,. (I) (E1,E2).. (A) I,E1 E2 I., κ.(b) I, E1 E2 I., E1 κ E2 κ. (C) I,E1 E2, I., κ. (D) I, E1 E2, I., E2 κ E1 κ. [59]. Fig 3 6 Shunting coefficient κ in branched dendrites measured in experiments. Data are from Ref. [59]. The data in grey was collected from 7 neurons and lines connect data from the same cell. The data in black is the average of the data in grey. In all figure panels, the locations of the inhibitory input (I) and excitatory inputs (E1 and E2) are marked by blue dot and red dots, respectively. The I path is marked by green. (A) The inhibitory input I at an oblique branch: κ is nearly constant for two distal E1 and E2 on the same branch. (B) As in (A) except that E1 and E2 are more proximal than I. κ is significantly different at E1 and E2 sites. (C)The inhibitory input I at the trunk: κ is nearly constant between E1 at the trunk and E2 at the oblique branch. (D) The inhibitory input I at an oblique branch: κ is significantly different between E1 and E2, where E1 is on the same branch as I and E2 is on a different branch. Figures are modified from Ref. [59]. 54

81 [59] (3 2)., (3 2) -, EPSP.,,. (3 2), (i), -, -, -, - ; (ii), EPSP ; (iii),.,. -, -, -,. : ,, ,, l, d. x. x = x E t = t E, x = x I t = t I, v c v t = g Lv q=e,i f q g q (t t q )δ(x x q )(v ε q ) + d 4r a 2 v x 2, (4 1) 55

82 v, c, r a, g L. f E f I. g E g I, ε E ε I ,,, v x =, x=l (4 2) v(, t) c = g L v(, t) + πd2 v t 4Sr a x, x= (4 3) S., v(x, ) =. (4 4), 3.2.2, t E t I., v(x, t) f E f I v(x, t) = k=m+n=k f m E f n I v mn (x, t). (4 5) (4 5) (6 1)-(6 1),, v mn (x, t). v mn (x, t) (m + n 2). O(1), O(f E ), v (x, t) =. (4 6) v 1 (x, t) = G(x, x E, t) [ε E g E (t t E )], (4 7). G(x, y, t), O(f 2 E ), v 2 (x, t) = G(x, x E, t) [ g E (t t E )v 1 (x E, t)], (4 8) O(f I ), v 1 (x, t) = G(x, x I, t) [ε I g I (t t I )], (4 9) 56

83 O(f 2 I ), O(f E f I ), v 2 (x, t) = G(x, x I, t) [ g I (t t I )v 1 (x I, t)]. (4 1) v 11 (x, t) = G(x, x E, t) [ g E (t t E )v 1 (x E, t)] + G(x, x I, t) [ g I (t t I )v 1 (x I, t)]. (4 11) ε E =7mV ε I = 1mV, (4 11) v 11 v 11 (x, t) G(x, x I, t) [ g I (t t I )v 1 (x I, t)]. (4 12) 3.2.4,,,, EPSP, V E (t) f E v 1 (, t) + fev 2 2 (, t). (4 13), IPSP, V I (t) f I v 1 (, t) + fi 2 v 2 (, t). (4 14), SSP, V S (t) V E (t) + V I (t) + f E f I v 11 (, t). (4 15) V SC (t) SSP EPSP+IPSP, V SC (t) f E f I v 11 (, t). (4 16) ε I mv, V I, V SC., V I,V SC, [59]., O(f E f I ). κ EI κ EI (t; t E, t I, x E, x I ) = V SC V E V I G(, x I, t) [g I (t t I )G(x I, x E, t) g E (t t E )] ε I G(, x E, t) g E (t t E ) G(, x I, t) g I (t t I ), (4 17a) (4 17b) 57

84 κ EI EPSP IPSP, f E f I (4 17b). (4 17a), V S (t) = V E (t) + V I (t) + κ EI (t)v E (t)v I (t). (4 18) κ EI EPSP IPSP., κ EI x E x I., κ EI t τ = t E t I, 4 1. (4 18) EPSP IPSP, (4 18) (4 18),, A, (t E = t I ), 4 2.A, x E = 283µm x I = 151µm, SSP EPSP IPSP., (4 18) EPSP t p., f E EPSP.5mV 6mV, f I IPSP -.5mV -3mV. f E f I, EPSP, IPSP SSP. f E f I, 9 { EPSP, IPSP, SSP }. t p SC V SC (t p ) EPSP IPSP, V E (t p )V I (t p ). 4 3.A, κ EI (t p ) EPSP IPSP. [59]., (4 18) t, t p. κ EI (t) t p τ < t < t p + τ, τ = 1ms. 58

85 κ EI (t,τ) E t I τ t E t 1µm I EPSP IPSP 4 1 κ EI. -, κ EI t τ... ( ) IPSP EPSP,. ( ) κ EI EPSP,. Fig 4 1 Shunting coefficient κ EI as a function of time t and input arrival difference τ for a fixed pair of excitatory and inhibitory input locations. Left, a morphological plot of a pyramidal neuron. The excitatory and inhibitory input locations are indicated by arrows. Right, (lower) an IPSP arrives at the soma earlier than an EPSP. The arrival times are indicated by vertical dashed lines. (upper) The shunting coefficient κ EI remains at zero until the time when the EPSP starts. 59

86 A EPSP Linear Sum B EPSP Linear Sum SSP SSP t p IPSP SC 5ms 2mV IPSP SC 5ms 2mV (A), EPSP, IPSP, SSP SC EPSP+IPSP. t p EPSP. (B) (A), IPSP EPSP 2ms x E = 283µm, x I = 151µm. Fig 4 2 The membrane potential profiles for a pair of concurrent and non-current E-I inputs. (A) An example of EPSP, IPSP, SSP, SC, and the corresponding linear sum when the EPSP and the IPSP are elicited concurrently. Here t p denotes the time when EPSP reaches its peak value. (B) The same as (A) except that the IPSP is elicited 2ms before the EPSP. The results are obtained in the realistic neuron model simulation which is described in detail in Section The excitatory input is given at the location x E = 283µm and the inhibitory input is given at the location x I = 151µm. 2ms, EPSP, EPSP,. t, κ EI (t) 9 { EPSP, IPSP, SSP }. 4 3.B, t, κ EI., R 2 1.,V SC (t) V E (t)v I (t). t,κ EI EPSP IPSP., t p,, EPSP IPSP mv., SC, (4 18) κ EI = mv 1. (t E t I ), 4 2.B, 2ms, (4 18), 4 4.A-B.,. 6

87 A V SC (mv) x V E V I (mv 2 ) B R 2 κ EI x Time (ms) (A) EPSP. (B) t p τ < t < t p + τ, τ = 1ms. ( ) V SC V E V I R 2. ( ) κ EI (t) ( mv 1 ),. 95%. (A). Fig 4 3 Numerical results for the dendritic integration of a pair of concurrent excitatory and inhibitory inputs. (A) Dendritic integration at the EPSP peak time. (B) Dendritic integration in the time interval t p τ < t < t p + τ, τ = 1ms. (upper) R 2 for the goodness of the linear fitting of V SC vs. V E V I at different times. (lower) The shunting coefficient κ EI (t) (in the unit of mv 1 ) as the slope of the linear fitting is plotted at different times. The error bar indicates 95% confidence interval. The circle marked by red indicates the case in (A) , (4 18). CA1., x E 1µm, x I 5µm., EPSP 1mV 8mV, IPSP -.5mV -3mV., 1ms, SC V SC EPSP IPSP V E V I, V E V I 1,.,. 61

88 A x1-1 B 6 1 R 2.5 V SC (mv) x1 κ EI V E V I (mv 2 ) Time (ms) IPSP EPSP 2ms. (A) EPSP. (B) t p τ < t < t p + τ, τ = 1ms. ( ) V SC V E V I R 2. ( ) κ EI (t) ( mv 1 ),. 95%. (A). Fig 4 4 Numerical results for the dendritic integration of a pair of nonconcurrent excitatory and inhibitory inputs. IPSP is elicited 2ms before the EPSP. (A) Dendritic integration at the EPSP peak time. (B) Dendritic integration in the time interval t p τ < t < t p + τ, τ = 1ms. (upper) R 2 for the goodness of the linear fitting of V SC vs. V E V I at different times. (lower) The shunting coefficient κ EI (t) (in the unit of mv 1 ) as the slope of the linear fitting is plotted at different times. The error bar indicates 95% confidence interval. The circle marked by red indicates the case in (A). (t E = t I ), t p SC V SC (t p ) EPSP IPSP, V E (t p )V I (t p ), 4 5.A. κ EI EPSP IPSP, (4 18). t p τ < t < t p + τ, τ = 1ms, V SC (t) V E (t)v I (t), 4 5.B. (4 18). (t E t I ), 2ms, (4 18) EPSP, 4 6.A-B., R , R (4 18). 62

89 A B 3 1 R 2.5 V SC (mv) 2 1 κ EI x V E V I (m V 2 ) Time (ms) (A) EPSP. (B) t p τ < t < t p + τ, τ = 1ms. ( ) V SC V E V I R 2. ( ) κ EI (t) ( mv 1 ),. 95%. (A). Fig 4 5 Experimental results for the dendritic integration of a pair of concurrent excitatory and inhibitory inputs. (A) Dendritic integration at the EPSP peak time. (B) Dendritic integration in the time interval t p τ < t < t p + τ, τ = 1ms. (upper) R 2 for the goodness of the linear fitting of V SC vs. V E V I at different times. (lower) The shunting coefficient κ EI (t) (in the unit of mv 1 ) as the slope of the linear fitting is plotted at different times. The error bar indicates 95% confidence interval. The circle marked by red indicates the case in (A) (4 18).., [26],,., x = x E1 t = t E1 f E1, x = x E2 t = t E2 f E2 63

90 A B 1 V SC (mv) 4 2 R κ EI x V E V I (mv 2 ) Time (ms) IPSP EPSP 2ms. (A) EPSP. (B) t p τ < t < t p + τ, τ = 1ms. ( ) V SC V E V I R 2. ( ) κ EI (t) ( mv 1 ),. 95%. (A). Fig 4 6 Experimental results for the dendritic integration of a pair of nonconcurrent excitatory and inhibitory inputs. IPSP is elicited 2ms before the EPSP. (A) Dendritic integration at the EPSP peak time. (B) Dendritic integration in the time interval t p τ < t < t p + τ, τ = 1ms. (upper) R 2 for the goodness of the linear fitting of V SC vs. V E V I at different times. (lower) The shunting coefficient κ EI (t) (in the unit of mv 1 ) as the slope of the linear fitting is plotted at different times. The error bar indicates 95% confidence interval. The circle marked by red indicates the case in (A)., c v t = g Lv q=e1,e2 f q g E (t t q )δ(x x q )(v ε E ) + d 2 v (4 19) 4r a x 2, 3.2.2,,, v x =, (4 2) x=l v(, t) c t = g L v(, t) + πd2 4Sr a 64 v x, (4 21) x=

91 , v(x, ) =. (4 22), (4 19) f E1 f E2, ( ) EPSP SSP, V S (t) = V E1 (t) + V E2 (t) + κ EE (t)v E1 (t)v E2 (t), (4 23) V E1 V E2 f E1 f E2 EPSP. V S SSP , κ EE (t), EPSP. κ EE - κ EI, (4 23),. (t E1 = t E2 ), x E1 f E1, EPSP 1. x E2 f E2, EPSP 2. x E1 x E2 f E1 f E2, SSP. f E1 f E2 EPSP 1 EPSP 2.5mV 2mV, 1 { EPSP 1, EPSP 2,SSP }., EPSP 1 t p, V SC (t p ) = V S (t p ) V E1 (t p ) V E2 (t p )., V SC V E1 (t p )V E2 (t p ), 4 7.A. κ EE EPSP 1 EPSP 2, (4 23) t p., 4 7.B, (4 23) t p τ < t < t p + τ, τ = 1ms. 65

92 A x B 1 R 2.5 V SC (mv) 1.5 x κ EE V E1 V E2 (m V 2 ) Time (ms) (A) EPSP. (B) t p τ < t < t p + τ, τ = 1ms. ( ) V SC V E1 V E2 R 2. ( ) κ EE (t) ( mv 1 ),. 95%. (A). Fig 4 7 Numerical results for the dendritic integration of a pair of concurrent excitatory inputs. (A) Dendritic integration at one of the EPSPs peak time. (B) Dendritic integration in the time interval t p τ < t < t p +τ, τ = 1ms. (upper) R 2 for the goodness of the linear fitting of V SC vs. V E1 V E2 at different times. (lower) The shunting coefficient κ EE (t) (in the unit of mv 1 ) as the slope of the linear fitting is plotted at different times. The error bar indicates 95% confidence interval. The circle marked by red indicates the case in (A). (t E1 t E2 ),, (4 23), 4 8., ( V E1 V E2 > 5mV), (4 23).., κ EE ( 1 2 ) κ EI ( 1 1 ). κ EE = mv 1 66

93 A x1-2 B 1 V SC (mv) κ R 2 EE x V E1 V E2 (m V 2 ) Time (ms) EPSP EPSP 2ms. (A) EPSP. (B) t p τ < t < t p + τ, τ = 1ms. ( ) V SC V E1 V E2 R 2. ( ) κ EE (t) ( mv 1 ),. 95%. (A). Fig 4 8 Numerical results for the dendritic integration of a pair of nonconcurrent excitatory inputs. One of the EPSPs is elicited 2ms earlier than the other. (A) Dendritic integration at one of the EPSPs peak time. (B) Dendritic integration in the time interval t p τ < t < t p + τ, τ = 1ms. (upper) R 2 for the goodness of the linear fitting of V SC vs. V E1 V E2 at different times. (lower) The shunting coefficient κ EE (t) (in the unit of mv 1 ) as the slope of the linear fitting is plotted at different times. The error bar indicates 95% confidence interval. The circle marked by red indicates the case in (A)., SSP EPSP. V S (t) = V E1 (t) + V E2 (t). (4 24) [88]..,,

94 A B SSP (mv) 4 SSP (mv) Linear Sum (mv) Linear Sum (mv) , (A) (B). (B) EPSP EPSP 2ms. x E1 5µm x E2 1µm. Fig 4 9 Dendritic integration of a pair of excitatory inputs in experiments. Our experimental result shows the nearly linear summation for a pair of concurrent excitatory inputs (A) and nonconcurrent excitatory inputs with arrival time difference 2ms (B). Two excitatory inputs are given at the location x E1 5µm and at x E2 1µm , x = x I1 t = t I1 f I1, x = x I2 t = t I2 f I2, c v t = g Lv q=i1,i2 f q g I (t t q )δ(x x q )(v ε I ) + d 2 v (4 25) 4r a x 2,,, v x =, (4 26) x=l 68

95 v(, t) c t = g L v(, t) + πd2 4Sr a, v x, (4 27) x= v(x, ) =. (4 28), (4 25) f I1 f I2, ( ) IPSP SSP, V S (t) = V I1 (t) + V I2 (t) + κ II (t)v I1 (t)v I2 (t), (4 29) V I1 V I2 f I1 f I2 IPSP. V S SSP , κ II (t), IPSP (4 29),. (t I1 = t I2 ),, x I1 = 94µm f I1, IPSP 1. x I2 = 151µm f I2, IPSP 2. x I1 x I2 f I1 f I2, SSP. f I1 f I2 IPSP 1 IPSP 2 -.5mV -3mV, 1 { IPSP 1,IPSP 2,SSP }., IPSP 1 t p, V SC (t p ) = V S (t p ) V I1 (t p ) V I2 (t p )., V SC V I1 (t p )V I2 (t p ), 4 1.A. κ II IPSP 1 IPSP 2, 69

96 A B 1 R V SC (mv) κ II V I1 V I2 (mv 2 ) Time (ms) (A) IPSP. (B) t p 5ms < t < t p + 15ms. ( ) V SC V I1 V I2 R 2. ( ) κ II (t) ( mv 1 ),. 95%. (A). Fig 4 1 Numerical results for the dendritic integration of a pair of concurrent inhibitory inputs. (A) Dendritic integration at one of the IPSPs peak time. (B) Dendritic integration in the time interval t p 5ms < t < t p + 15ms. (upper) R 2 for the goodness of the linear fitting of V SC vs. V I1 V I2 at different times. (lower) The shunting coefficient κ II (t) (in the unit of mv 1 ) as the slope of the linear fitting is plotted at different times. The error bar indicates 95% confidence interval. The circle marked by red indicates the case in (A). t p., 4 1.B, (4 29) t p 5ms < t < t p + 15ms., (4 29), , x I1 5µm, x I2 1µm., IPSP 1 IPSP 2 -.5mV -3.5mV. (t I1 = t I2 ), t p SC V SC (t p ) 7

97 A B 1.5 R V SC (mv) 1.5 κ II V I1 V I2 (mv 2 ) Time (ms) IPSP IPSP 2ms. (A) IPSP. (B) t p 5ms < t < t p +15ms. ( ) V SC V I1 V I2 R 2. ( ) κ II (t) ( mv 1 ),. 95%. (A). Fig 4 11 Numerical results for the dendritic integration of a pair of nonconcurrent inhibitory inputs. One of the IPSPs is elicited 2ms earlier than the other. (A) Dendritic integration at one of the IPSPs peak time. (B) Dendritic integration in the time interval t p 5ms < t < t p + 15ms. (upper) R 2 for the goodness of the linear fitting of V SC vs. V I1 V I2 at different times. (lower) The shunting coefficient κ II (t) (in the unit of mv 1 ) as the slope of the linear fitting is plotted at different times. The error bar indicates 95% confidence interval. The circle marked by red indicates the case in (A). IPSP 1 IPSP 2, V I1 (t p )V I2 (t p ), 4 12.A. κ II IPSP 1 IPSP 2, (4 29). t p 5ms < t < t p + 15ms, V SC (t) V I1 (t)v I2 (t), 4 12.B. (4 29). (t I1 t I2 ), 2ms, (4 29) IPSP,

98 A B R 2.5 V SC (mv) κ II V I1V I2 (mv 2 ) Time (ms) (A) IPSP. (B) t p 5ms < t < t p + 15ms. ( ) V SC V I1 V I2 R 2. ( ) κ II (t) ( mv 1 ),. 95%. (A). Fig 4 12 Experimental results for the dendritic integration of a pair of concurrent inhibitory inputs. (A) Dendritic integration at one of the IPSPs peak time. (B) Dendritic integration in the time interval t p 5ms < t < t p + 15ms. (upper) R 2 for the goodness of the linear fitting of V SC vs. V I1 V I2 at different times. (lower) The shunting coefficient κ II (t) (in the unit of mv 1 ) as the slope of the linear fitting is plotted at different times. The error bar indicates 95% confidence interval. The circle marked by red indicates the case in (A) ,., [28].., c v t = g Lv q=e,i G q (v ε q ) + d 4r a 2 v x 2, (4 3) 72

99 A 2. B R 2.5 V SC (mv) κ II V I1 V I2 (mv 2 ) Time (ms) IPSP IPSP 2ms. (A) IPSP. (B) t p 5ms < t < t p +15ms. ( ) V SC V I1 V I2 R 2. ( ) κ II (t) ( mv 1 ),. 95%. (A). Fig 4 13 Experimental results for the dendritic integration of a pair of nonconcurrent inhibitory inputs. One of the IPSPs is elicited 2ms earlier than the other. (A) Dendritic integration at one of the IPSPs peak time. (B) Dendritic integration in the time interval t p 5ms < t < t p + 15ms. (upper) R 2 for the goodness of the linear fitting of V SC vs. V I1 V I2 at different times. (lower) The shunting coefficient κ II (t) (in the unit of mv 1 ) as the slope of the linear fitting is plotted at different times. The error bar indicates 95% confidence interval. The circle marked by red indicates the case in (A). G q = M q i=1 j=1 f ij q g q (t t ij q )δ(x x i q), (4 31) q {E, I}. q, fq ij ith jth, t ij q ith jth, x i q ith., SSP 73