**growing**with time while a constant current was inserted. When the current injection stopped, the voltage gradually attenuated with time to zero. So a cell can't be represented as a mere resistance because if the cell behavior would have been like a resistor then the Voltage should have been equal to V=I.R (

*Ohm's law, here*) and looking something like this:

**grows with time**and this is a typical reaction of any cell if one injects small positive current. This sort of "growth" (or better "the positive voltage

**response**to the injection of positive current") is called

**depolarization**, it's a depolarized current. So the simplest circuit that is able to represent such a voltage behavior, as a first approximation, is the

**RC circuit**(

*see here and here and here. You may also need to understand what a resistor and a capacitor is in the first place*).

**iso-potential cell**meaning that there is no voltage change (no drop of voltage) across its membrane and the voltage difference between inside and outside of it, at any point of the membrane will be V. So to measure V in response to injected I, we can write the following equation:

*see Kirchhoff's law*, here). According to this law (between other things) the current inserted in an RC circuit must split. So the capacitative current (C (=dV/dt)) plus the resistive current (I=V/R (->Ohm's law) must equal to the current I, the injected current. Now, if you charge the capacitor with positive current, then the inside of the cell gets positive (something that is also depicted by the gradual, upward movement of voltage but it's also evident in the schematics of the RC circuit above).

*and we want to do this in order to understand what's going on with the voltage in the cells of our brain*) one must define initial conditions. So we already said that we start with V(t=0)=0, meaning that before we do anything (t=0), the voltage equals to zero (we'll soon see that the "before we do anything" expression is better called the "resting state" of a brain cell and also that the aforementioned zero number is not actually zero volts but -70mV (minivolts)). So the solution looks like this:

1. Voltage V as a function of time (V(t)) is equal to injected current I multiplied by R (which is the resistance) multiplied by 1 minus "e" (exponent) in the power of minus "t" (time) divided by R multiplied by C.

2. Let's check the equation by assuming the time (t) is indeed zero (meaning that we're at the "initial condition" (the "resting state", see below). So if t=o, then "e" to the power of zero is 1, 1 minus 1 is zero, so V(t=0)=0. This means that our "initial condition" (t=0) is satisfied.

3. Now let's look at the other extreme which is the case where time (t) equals infinity (∞), meaning that I inject my current for a very long time. So if (t) goes to infinity then "e" to the power of (t) goes to zero, then 1 minus zero is 1 and finally you're left with V(t=∞)=I.R which means that the cell/circuit enters the "steady state" (

*check the capacitors theory video above for explanation of the steady state,*--ok, don't scroll up, just click here) or better, the voltage remains at its steady state.

*Okay, and now for a short break. Here is one of my super-ultra favorite ABBA reprises by At Vance, a bunch of some excellent metallers from Germany. Your brain cells will love it! (Listen first, judge next!!)*

*In Greek, we pronounce this letter like /taf/ and not like /tau/ , for what it's worth, never mind, I understand it's a universal scientific terminology*). "Tau" is a unit of time calculated in units of seconds. So let's see what happens to voltage when you inject t=R.C=τ. Check screenshot below:

*arbitrary for now - not explained*) equal to 0.63

*(explanation from a co-student and reader of this blog (check his/her comment below): 'e' is a constant, approximately equal to 2.72. Raised to the power -1 gives 0.37, and subtracted from 1 finally gives 0.63*). So this means that once time (t) equals "tau", I get a voltage increase that gets into 63% of the maximum value it can get.

**exponentially (1 minus exponent)**and eventually if I wait enough it will get asymptotically close to its maximum value. Now let's see how the voltage decays after the current stops.

**electrical memory behavior**because it takes time to read the previous state and discharge the capacitance and get rid of the injected current. Short τm develops fast voltage and gets rid of it and long τm takes long to charge and discharge. Also we can now tell that the initial schema of the RC circuit is a

**passive (resting) system**, meaning it's values are linear and fixed and do not change during the injection. Finally, also R is a significant parameter since it tells us how much voltage I get (V=I.R), or how much depolarization I get. If the resistance is big, then I get big voltage (and vice versa). Now let's move to Temporal Summation.

**adds to it**the new amount of injected current on

**top of the remainders of the previous one.**So in other words, when you have a sequence of several inputs with appropriate time difference, then the remainders of the previous conditions wll become the initial condition for the next input to build-up.

**this means that when you have consequent synaptic inputs between cells, the voltage of them will sum up one of top to another, thus letting afterwards the AIS region of the axon to decide if a certain voltage threshold has been reached in order to generate (or not) a spike.**The interplay between positive and negative current, meaning current that will either try to depolarize or hyperpolarize the cell, is exactly what synapses are doing.

*The temporal summation then will be the total sum between the interplay of excitatory cells and inhibitory ones*. The way a neuron works when it sums inputs is governed by the temporal summation principle which is in turn the result of the time constant or the electric memory of the cell. Let's move on to the Resting Potential (mentioned at some place above).

**The mathematical equation of the full post-synaptic cell:**so now we have three types of current in our new type of circuit. The resistive current (passive), the capacitative current and the new synaptic current. The sum of all currents, according to Kirchhoff's law should equal zero

That was all! Keep it up, beginner neuroscientist, you!

**Some general electricity notes**

(

*source: khanacademy, brightstorm @ YouTube)*

1. Ohm’ s Law: Current through a circuit is proportional to voltage (I=V/R). V is calculated with volts, R with Ω (ohms) and I with A (amperes, which in turn equal to C/S (=coulomb/second)). We could also say that V=I.R. A circuit is a “bunch” of wires that connect a set of circuit elements.

2. Current is the flow of charge per second, or the change of charge (ΔQ) to change of time (sec. or Δt).

3. The resistor determines the rate with which the electrons travel through the entire circuit (or all electrons travel with the same speed in a given circuit). The resistor serves as a “bottleneck” which regulates the speed of current. Resistors use energy.

4. In traditional/conventional schematics, the current flows from positive to negative pole in a circuit but this is wrong. In reality, it flows the other way.

5. The amount of voltage (or potential difference) in a circuit determines the “urge” or the “speed” with which the electrons (negative pole) want to travel and get to the positive pole.

6. Voltage is not an absolute number. It is a calculated number, it’s the potential difference between two points in a circuit and this number is always constant.

7. Resistors in series increase the probability of an electron “bump” onto something so it decelerates (and creates heat). The total resistance in such a circuit equals to the sum of each resistor. So resistors in series increase the total resistance of a circuit (example: Christmas lights, in goes out, all go out). Also in resistor in-series, the current (I ) is the same.

8. Resistors in parallel create “branches” within the circuit. The current (Amperes) through each branch is proportional to the resistance in Ω’s of each resistor. To find the total resistance (equivalent resistance) between resistors in parallel equals to: 1/Rt=1/R1+1/R2+1/Rn. So resistors in parallel reduce the overall resistance and the smallest current flows through the largest resistor (since it creates bigger resistance so less current can flow through it). So the effective resistance (or total resistance of a circuit) of resistors in parallel is always smaller than any of the resistors in parallel (example: house wiring, all electrical devices are separate parallel branches of the same circuit, since once doesn’t want to turn the TV in order to turn on the oven). In resistors in-parallel the voltage (or potential difference – V) is the same.

9. A circuit is a “bunch” of wires that connect a set of circuit elements.

10. Power (wattage) equals to amperes (current) times volt (voltage), (P=I.V) and is the rate of energy use. P also equals to V2/R since I=V/R or I2.R since V=I.R.