GNU General Public License v3.0 licensed. Source available on github.com/zifeo/EPFL.
Sping 2017: Biological Modeling of Neural Networks
[TOC]
- cortex : frontal, motor, visual
- neuron : 10K per cubic cm, 3km wire
- spike : rare events triggered at threshold, action potential
- reset : refactoriness
- postsynaptic potential :
$\epsilon_{ij}(t)=u_i(t)-u_{rest}$ - excitatory : EPSP
- inhibitory : IPSP
- membrane potential : fluctuating
- passive membrane model :
$\tau\ddt u = -(u-u_{rest})+RI(t)$ - circuit : capacitor in parallel with resistance,
$I=I_R+I_C$ with$I_C=C\ddt u$ and$u=RI$ - free solution :
$u(t)=u_{rest}+\int_{-\infty}^\infty\frac{1}{C}e^{-(t-t')/\tau}I(t')dt'$ - single pulse :
$\delta u(t)=\frac{q}{C}e^{-(t-t_0)/\tau}$
- circuit : capacitor in parallel with resistance,
- pulse current input :
$I(t)=q\cdot \delta(t-t_0)$ - leaky integrate and fire model LIF :
$\tau\ddt u = -(u-u_{rest})+RI(t)$ with fire at$u(t)=\vartheta$ and resetß$u\to u_{reset}$ 
- frequency-current
- frequency-current
- nonlinear integrate and fire NLIF :
$\tau \ddt u=F(u)+RI(t)$ 
- quadratic :
$F(u)=c_2(u-c_1)^2+c_0$ - exponential :
$F(u)=-(u-u_{rest})+c_0 e^{u-\vartheta}$
- quadratic :
- resting potential :
$-70mV$
- membrane
- ion channels :
$C\ddt u=-\sum_k I_{ion,k}+I(t)$ , 200 identified ones, open/close stochasticly - ion pumps : concentration difference giving voltage difference
- inside
$n_1$ :$K$ potassium - outside
$n_2$ :$Na$ sodium - probability to be in a state with energy
$E$ :$n\propto e^{-\frac{E}{kT}}$ with Boltzmann constant$kT$
- ion channels :
- Hodgin-Huxley : 4 differential equation, 4D, threshold depends on stimulus, voltage threshold good approximation
- f-I curve
- constant current input
- pulse input
- step current
- constant current input
- threshold : not strict, coupled differential equations, refactoriness
- f-I curve
- reversal potential : $\Delta u = u_1-u_2=-\frac{kT}{q}\ln\frac{n(u_1)}{n(u_2)}$no ion flow, compute by nernst equation
- reduce HH model to 2 dimensions :
$C\ddt u=-g_{Na}m_0(u)^3(1-w)(u-E_{Na})-g_K(\frac{w}{a})^4(u-E_K)-g_l(u-E_l)+I(t)$ with$\ddt w=-\frac{w -w_0(u)}{\tau_{eff}(u)}$ - separation of time scales for
$\tau_1 <<\tau_2$ : dynamics of$m$ fast,$m(t)=m_0(u(t))$ - exploit similarities/correlations : dynamics of
$h$ and$n$ similar,$1-h(t)=an(t)=w(t)$ - assume
$I(t)$ slow
- separation of time scales for
- 2 dimensional equations :
$C\ddt u=f(u(t),w(t))+I(t)$ with$\ddt w=g(u(t),w(t))$ -
$\tau_1<<\tau_2$ :$x=h(y)$ approximate$\tau_1\ddt x=-x+h(y)$ - simplify :
$\tau_2\ddt y=f(y)+g(h(y))$
-
- phase plane analysis : flow arrow
- nullcline : all points with
$\ddt u=0$ or$\ddt w=0$ - consider : change in small time step
- stable fixed point : characterzied by Eigenvalues
- linearized equations $\ddt x=\begin{pmatrix}F_u& F_w\\ G_u& G_w\end{pmatrix}x$
- solution form :
$x(t)=e^{\lambda t}$ - two solutions :
$\lambda_++\lambda_-=F_u+G_w$ and$\lambda_+\lambda_-=F_uG_w-F_wG_u$ - stability :
$\lambda_+ < 0$ and$\lambda_- < 0$ - implication :
$F_u+G_w <0$ and$F_uG_w-F_wG_u > 0$ 
- intersection : moves and changes stability
- nullcline : all points with
- FitzHugh-Nagumo model
- type I and II model
- glutamate : neurotransmitter at excitatory synapses,
$E_{syn}\approx 0mV$ - AMPA channel : rapid, calcium cannot pass if open
- NMDA channel : slow, calcium can pass if open
- GABA : neurotransmitter at inhibitory synapses,
$E_{syn}\approx -75mV$ - GABA-A
- GABA-B
- basic model
- time rise model
- synpatic short-term plasticity
- change induced : over 0.5s
- recover : over 1s
- depression
- faciliation
- dendrite : longitudinal resistance
- cable equation
- passive dendrite : leak
- active dendrite : Ca, Na
- axon : Na, K
- passive dendrite : leak
- compartmental models : many ion channels, spatially distributed, morphologically reconstructed, difficult to tune
- backpropagating action potential BPAP
- limit cycle : oscillation
- containing one unstable fixed point
- no other fixed point
- bounding box with inward flow
- type II model : discontinuous gain function
- Hopf bifuraction : stability loss give oscillation with finite frequency
- Hopf bifuraction : stability loss give oscillation with finite frequency
- type I model : size of arrows matters
- saddle-node bifurcation
- stable/unstable manifold : attractor/repellor
- delayed spike
- separation of time scales :
$\tau_w >>\tau_u\to\Delta w<<\Delta u$ 
- reduction to 1 dimension :
$\tau_w >>\tau_u\to w\approx w_{rest}$ 
- brain
- distributed architecture :
$10^{10}$ neurons - connections/neurons :
$10^4$ - memory separation from processing : no
- distributed architecture :
- associative memory : pattern completion, word recognition
- classification by similarity : closest prototype
$\abs{x-p^T}\le\abs{x-p^A}$ - magnetic materials :
$S_i=\pm 1$ ,$w_{ij}=\pm 1$ (same state or not) with$S_i(t+1)=sgn[\sum_j w_{ij}S_j(t)]$ over all interactions- noisy magnet : arrow missaligned
- pure magnet : everything aligned
- hopfield model : maximum overlap similarity, random patterns, fully connected,
$w_{ij}=\sum_\mu p_i^\mu p_j^\mu$ - update :
$S_i(t+1)=sgn[h_i(t)]=sgn[\sum_j w_{ij} S_j(t)]$ - overlap :
$m^\mu(t)=\frac{1}{N}\sum_j p_j^\mu S_j(t)$
- update :
- hebbian learning : local rule, simultaneously active (correlation), fire together wire together
- Stroop effect : hard to work against natural associations
- storage capacity : random walk
- minimal condition : pattern fixed point of dynamics
- retrieval requires more
- hopfield model : for small number of patterns, states with overlap 1 are fixed point
- stochastic hopfield model :
$P(S_i(t+1)=+1\mid h_i)=g[h_i]=g[\sum_j w_{ij} S_j(t)]=g[\sum_\mu p_i^\mu m^\mu(t)]$ -
$g(h_i)=0.5[1+\tanh(2h)]$ 
-
- energy landscape :
$E=-\frac{1}{2}\sum_{i,j}w_{ij}S_iS_j$ - random pattern vs orthogonal patterns
- attractor networks : dynamics moves network state to fixed point,
$w_{ij}=\frac{1}{N}\sum_\mu (p_i^\mu - b)(p_j^\mu-a)$ with$a,b$ means of activity- inhibition/excitation separated : active together
$w_{ij}=\frac{1}{N}\sum_\mu(p_i^\mu+1)(p_j^\mu+1)$
- inhibition/excitation separated : active together
- rate model : active = high rate = many spikes per second
- hebbian model :
$\ddt w_{ij}=F(w_{ij};v_j^{pre},v_i^{post})=a_0+a_1^{pre}v_j^{pre}+a_1^{post}v_i^{post}+a_2^{corr}v_j^{pre}v_i^{post}+\cdots$ 
- BCM rule
- BCM rule
- plasticity : memory formation/retention/stability
- homosynaptic/Hebb : pre and post
- heterosynapitc : post, can self-stabilize
- transmitter-induced : pre
- long-term persistance : LTP vsLTD
- population activity :
$A(t)=\frac{n(t;t+\Delta t)}{N\Delta t}$ , similar properties - cortical column : neighboring neurons has similar properties
- receptive field : retina, LGN (surrounding), visual cortex (orientation selective), spatially localized
- neuronal population : group of neurons with similar properties, input, receptive field, connectivity
- connectivity
- all-to-all
- random with number
$K$ input fixed - random with probability
- meanfield argument : all neurons receive the same total input current
- asynchronous state :
$<A(t)>=A_0=constant$ - fully-connected network : static coupling
$w_{ij}=w_0$ - pulse :
$\alpha$ - current :
$I(t)=I^{ext}(t)+I^{net}(t)$ with$I^{net}(t)=\sum_j\sum_f w_{ij}\alpha(t-t_j^f)$ and$w_{ij}=\frac{J_0}{N}$ - neuron independent :
$I(t)=J_0\int \alpha(s)A(t-s)ds+I^{ext}(t)$ - all variables constant in time :
$I_0=J_0qA_0+I_0^{ext}$ with$q=\int\alpha(s)ds$
- neuron independent :
- gain-function :
$\nu=g(I_0)$ , frequency current relationship - single neuron rate = population rate :
$\nu=A_0$ 
- asynchronous state :
- stochastic meanfield : when increase network size
- with
$p$ fixed :$w_{ij}=\frac{w_0}{pN}$ , fluctuation of$A$ decrease, fluctuation of$I$ decrease - with
$K$ connections : fluctuation of$A$ decrease, fluctuation of$I$ remain - balanced, fixed
$p$ : fluctuation of$A$ decrease, fluctuation of$I$ become smooth
- with
- stationary state
- single neuron = population rate of homogenous population
- activity predicted by gain function, external input and intra-population coupling strengh
- choice of neuron model irrelevant apart from gain
- transients : beyond stationary states, neuron model matters
- SRM and GLM
- uncoupled population
- low noise : fast transient, oscillation,
$A(t)\approx g(I(t))\approx\tilde{g}(I(t),I'(t))$ - high noise : slow transient,
$A(t)=F(h(t))$
- low noise : fast transient, oscillation,
- 1 population = 1 differential equation :
$\tau\ddt h(t)=-h(t)+RI^{ext}(t)+\gamma F(h(t))$ as$I(t)=I^{ext}(t)+J_0qF(h(t))$
- uncoupled population
- coarse coding : many cells respond to single stimulus with different rate, discrete
- spatial continuum :
$I(x,t)=I^{ext}(x,t)+d\int w(x-x',t)A(x',t)dx'$ and$\tau\ddt h(x,t)=-h(x,t)+RI(x,t)$ - integro-differential equation :
$\tau\ddt h(x,t)=-h(x,t)+RI^{ext}(x,t)+d\int w(x-x')F(h(x',t))dx'$ - coupling across continuum : mexican hat, gaussian-like with elbows
- bump solution : activity profile in absence of input, strong lateral connectivity, current orientation
- uniform solution : edge enhancement effect, matchband, contrast enhancement
- integro-differential equation :
- receptive fields dependence : direction of motion
$\beta$ with preferred direction$P$ (neutral direction$N$ )- coherence : correlation agreement between these neurons
- LIP neuron : selective to target of saccade, increase faster if signal strong, activity noisy
- decision dynamics : assumption F-I curve linear in some limited regime and inhibit population is fast (
$\tau_{inb}<<\tau_{exc}$ )
-
$A_{inh}(t)$ phase plane- strong external input
- biased input
$h_2^{ext}<h_1^{ext}$ : stable fixed point, decision reflects bias - symmetric but small input : stable fixed point
- symmetric but strong input : 2 stable fixed point
- homogeneous : saddle point, decision must be taken
-
- good neuron models
- predict spike time and subtreshold voltage
- easy to interpret
- flexible enough to account for variety of phenomena
- systematic procedure to optimize parameters
- best choice of
$f$ : linear + exponential$\tau\ddt u =-(u-u_{rest})+\Delta e^{\frac{u-\vartheta}{\Delta}}$ 
- need to add
- adaptation on slower time scale
- diversity of firing pattern
- increase threshold
$\vartheta$ after each spike : dynamic threshold :$/vartheta=\theta_0+\sum_f\theta_1(t-t^f)$ - noise
- AdEx model : adaptation and fire patterns
- $a$ : slow of$w$ -nullcline -$b$ :$w$ jump amount after spike
- need to add
- adaptive leaky integrate and fire
- spike response model : SRM, firing if
$u(t)=\vartheta(t)$ 
- espace noise :
$p(t)=p_0e^{\frac{u(t)-\vartheta}{\Delta}}$ 
- likelihood of spike train
- SRM with escape noise : GLM, model of encoding and decoding (predict stimulus, neuroprosthetics)
- espace noise :
- fitting models to data
- voltage/subthreshold : linear in parameters, quadratic error function
- spike times : nonlinear but GLM, convex error function
- limited in vitro
- how long does spike effect last : powerlaw
- variability : spike timing, membrane potential
- in vivo : data looks noisy
- in vitro : fluctuation
- source
- intrinsic noise : finite number of channels, finite temperature, stochastic opening and closing, small contribution, fairly reliable
- network noise : spike arrival, beyond of control of experimentalist, big contributions
- poisson model
- homogeneous : constant rate
- firing probability :
$P_F=p_0\Delta t$ - survivor function :
$\ddt S(t_1\mid t_0)=-p_0 S(t_1\mid t_0)$
- firing probability :
- inhomogeneous : rate changes, consistent with rate coding
- firing probability :
$P_F=p(t)\Delta t$ - survivor function :
$S(t\mid\hat t)=e^{-\int_{\hat t}^t p(t')dt'}$ - interval distribution :
$P(t\mid\hat t)=p(t)e^{-\int_{\hat t}^t p(t')dt'}$
- firing probability :
- homogeneous : constant rate
- rate codes
- temporal averaging :
$v(t)=\frac{n^{sp}}{T}$ single neuron, single trial, too slow for animal- interspike intervals : regularity, ISI broad distribution
- fano factor :
$F=\frac{<(n_k^{sp}-<n_k^{sp}>)^2>}{<n_k^{sp}>}$ , repeatability across repetitions
- averaging across repetitions : single neuron, many trial, not useful for animal,
$PSTH(t)=\frac{n(t;t+\Delta t)}{K\Delta t}$ - population averaging (spatial) : many neuron, natural readout,
$A(t)=\frac{n(t;t+\Delta t)}{N\Delta t}$
- temporal averaging :
- stochastic spike arrival : poisson model
- total spike train of
$K$ presynaptic neurons :$P_F=Kp_0\Delta t$ , take expectation$S(t)=\sum_{k=1}^K\sum_f\delta(t-t_k^f)$ as$\Delta t\to 0$ 
- passive membrane : LIF without threshold, can analytically predict mean of membrane potential fluctuation,
$I^{syn}(t)=I_0+I^{fluct}(t)$ - white noise :
$<\zeta(t)>=0$ , autocorrelation$<\zeta(t)\zeta(t')>=a^2\tau\delta(t-t')$ 
- discrete poisson : probability of firing
$p=v\Delta t$ with$\Delta t$ small time step, autocorrelation$<S(t)S(t')>=v_0\delta(t-t')+[v_0]^2$
- total spike train of
- noisy integrate and fire : passive membrane
$u(t)=\sum_k w_k\sum_f \epsilon(t'-t_k^f)$ - espace noise : rate
$p(t)=f(u(t)-\vartheta)=\frac{c}{\Delta}e^{\frac{u(t)-\vartheta}{\Delta}}$ ,$\tau\ddt u_i=-u_i + RI+\zeta(t)$ - interspike interval distribution : time-dependent,
$\ddt S_I(t\mid\hat t)=-p(t)S_I(t\mid\hat t)$ - interval distribution :
$S_I(t\mid\hat t)=e^{-\int_{\hat t}^t p(t')dt'}$ - survivor function :
$P_I(t\mid\hat t)=p(t)e^{-\int_{\hat t}^t p(t')dt'}$
- interval distribution :
- synapse plasticity : adapt statistics of task (receptive field), memorize facts, avoid blowup of activity, avoid useless loss of energy
- hemeostatis : activity control, avoid blow-up
- hebbian learning : unsupervised learning, statistical change, development (wiring receptive field)
- eligibility trace : synapse keep memory of pre-post hebbian learning
- reinforcement learning : success = reward - expectect reward, reward + hebbian,
$\Delta w_{ij}\propto F(pre,post,success)$ , useful new behavior- dopamine : reward/success
- neuromodulator : interestingness, surpise, attention, novelty, near-global action
- STDP model :
$\tau_+ \ddt z_j^+=-z_j^++\sum_f \delta(t-t_k^f)$ and$\tau_-\ddt z_i^-=-z_i^-+\sum_f\delta(t-t_i^f)$ giving$\Delta w_{ij}=\sum_{t_i^f}\sum_{t_j^f}W(t_j^f-t_i^f)$ 
- STDP to rate model :
$v_i^{post}(t)=\sum_j w_{ij}\sum_f \epsilon(t-t_j^f)$ 
- STDP triplet : enable to fit experimental data with rate based evolution
- pre-post :
$\ddt w_k=-Bz_i^-\delta(t-t_j^{pre})$ - post-pre-post :
$\ddt w_j=+A^+z_j^+z_i^{slow}\delta(t-t_i^{post})$ - assume poisson independent :
$\ddt w_j=-\alpha Bv_i^{post}v_j^{pre}+\beta A^+(v_i^{post})^2v_j^{pre}$
- pre-post :
- self stabilitizing rate based plasticity :
$\ddt w_{ij}=\cdots$ - nonlinear Hebb for potentiation :
$+a_3^{BCM}v_j^{pre}(v_i^{post})^2$ - pre-post for depression :
$-a_2^{LTD}v_j^{pre}v_i^{post}$ - heterosynaptic plasticity : pure post,
$-a_4^{het}(w_{ij}-z_{ij})[v_i^{post}]^4$ - transmitter-induced : pure pre,
$+a_1^{pre}v_j^{pre}$
- nonlinear Hebb for potentiation :